FOR IF THE BLOOD OF GOATS AND BULLS AND THE ASHES OF A HEIFER: ei gar to haima tragon kai tauron  kai spodos damaleos:  (Leviticus 16:14,16) (Numbers 19:2-21)

Leviticus 16:14 "Moreover, he shall take some of the blood of the bull and sprinkle it with his finger on the mercy seat on the east side; also in front of the mercy seat he shall sprinkle some of the blood with his finger seven times... 16 And he shall make atonement for the holy place, because of the impurities of the sons of Israel, and because of their transgressions, in regard to all their sins; and thus he shall do for the tent of meeting which abides with them in the midst of their impurities.

If here is a so-called first-class conditional statement (see note) and thus is assumed as true. The divinely ordained OT ceremonial rituals did indeed serve "for the (external) cleansing of the flesh" but it was not sufficient or efficacious for (internal) cleansing of one's conscience.

Not all the blood of beasts on Jewish altars slain,
Could give the guilty conscience peace or wash away the stain.
Christ the heavenly Lamb takes all our sins away,
A sacrifice of nobler name and richer big than they.
      -- Isaac Watts (Play entire hymn)

Cole explains that...

This was a ritual for purification, especially if someone had been defiled by touching a dead body. The author argues from the lesser to the greater. If these rituals could cleanse the flesh, “how much more will the blood of Christ, who through the eternal Spirit offered Himself without blemish to God, cleanse your conscience from dead works to serve the living God?” Jesus Christ is the only one who could atone for man’s sin, because He alone was a man without blemish in all that He did. Thus His blood can act as the substitute for the penalty that we deserve. (Hebrews 9:1-14 God's Remedy for Guilt)

AND THE ASHES OF A HEIFER SPRINKLING THOSE WHO HAVE BEEN DEFILED: kai spodos damaleos rantizousa (PAPFSN) tous kekoinomenous (RPPMPA):

Sprinkling (4472) (rhantizo) was used in secular Greek to describe common sprinkling in a non-religious sense but there were uses in which sprinkling conveyed the idea of religious cleansing.

Robertson explains that the ashes of the red heifer mingled with water...

were sprinkled on the contaminated or defiled ones (see Numbers 19) as (in the same way and for much the same purpose as) the blood of bulls and goats was offered for sins (on the Day of Atonement in Leviticus 16).

The picture of sprinkling reflects Jewish ritual purification in which the Jews would wash their hands before eating and then sprinkle themselves (rhantizo) with water when they return from the market place. The sprinkling was not upon objects bought in the market but persons who may have contracted defilement in mixing with others.

Defiled (2840) (koinoo from koinos = common, defiled, unclean, unholy, profane, that which lies common or open to all) means to make common or unclean, to profane, to pollute, to cause to become ritually unacceptable.

Wuest writes...

The writer in this verse speaks of the unclean Israelite, the person who was rendered ceremonially unclean by contact with a dead body, or by entering a house where a corpse was lying, or by touching a bone or a tomb. If he should enter the tabernacle while thus defiled, he was cut off from Israel. Ceremonial defilement was not in itself sin, but a type of sin. Hence the blood of animals could cleanse away this defilement. It was only the flesh of the person which was defiled by contact with the dead. It was likewise only the flesh that was cleansed. Thus, defilement and cleansing were both symbolic. (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Eerdmans or Logos) (Bolding added)

Moses explains the Red Heifer in Numbers 19 writing that...

Nu 19:2-21 "This is the statute of the law which the LORD has commanded, saying, 'Speak to the sons of Israel that they bring you an unblemished Red Heifer in which is no defect, and on which a yoke has never been placed. 3 'And you shall give it to Eleazar the priest, and it shall be brought outside the camp and be slaughtered in his presence. 4 'Next Eleazar the priest shall take some of its blood with his finger, and sprinkle some of its blood toward the front of the tent of meeting seven times. 5 'Then the heifer shall be burned in his sight; its hide and its flesh and its blood, with its refuse, shall be burned. 6 'And the priest shall take cedar wood and hyssop and scarlet material, and cast it into the midst of the burning heifer.

7 'The priest shall then wash his clothes and bathe his body in water, and afterward come into the camp, but the priest shall be unclean until evening. 8 'The one who burns it shall also wash his clothes in water and bathe his body in water, and shall be unclean until evening.

9 'Now a man who is clean shall gather up the ashes of the heifer and deposit them outside the camp in a clean place, and the congregation of the sons of Israel shall keep it as water to remove impurity; it is purification from sin. 10 'And the one who gathers the ashes of the heifer shall wash his clothes and be unclean until evening; and it shall be a perpetual statute to the sons of Israel and to the alien who sojourns among them.

11 'The one who touches the corpse of any person shall be unclean for seven days. 12 'That one shall purify himself from uncleanness with the water on the third day and on the seventh day, and then he shall be clean; but if he does not purify himself on the third day and on the seventh day, he shall not be clean.

13 'Anyone who touches a corpse, the body of a man who has died, and does not purify himself, defiles the tabernacle of the LORD; and that person shall be cut off from Israel. Because the water for impurity was not sprinkled on him, he shall be unclean; his uncleanness is still on him.

14 'This is the law when a man dies in a tent: everyone who comes into the tent and everyone who is in the tent shall be unclean for seven days. 15 'And every open vessel, which has no covering tied down on it, shall be unclean. 16 'Also, anyone who in the open field touches one who has been slain with a sword or who has died naturally, or a human bone or a grave, shall be unclean for seven days. 17 'Then for the unclean person they shall take some of the ashes of the burnt purification from sin and flowing water shall be added to them in a vessel. 18 'And a clean person shall take hyssop and dip it in the water, and sprinkle it on the tent and on all the furnishings and on the persons who were there, and on the one who touched the bone or the one slain or the one dying naturally or the grave. 19 'Then the clean person shall sprinkle on the unclean on the third day and on the seventh day; and on the seventh day he shall purify him from uncleanness, and he shall wash his clothes and bathe himself in water and shall be clean by evening. 20 'But the man who is unclean and does not purify himself from uncleanness, that person shall be cut off from the midst of the assembly, because he has defiled the sanctuary of the LORD; the water for impurity has not been sprinkled on him, he is unclean.

21 'So it shall be a perpetual statute for them. And he who sprinkles the water for impurity shall wash his clothes, and he who touches the water for impurity shall be unclean until evening.

The ashes were regarded as a concentration of the essential properties of the sin-offering, and could be resorted to at all times with comparatively little trouble and no loss of time. One red heifer availed for centuries. Only six are said to have been required during the whole of Jewish history because only a small quantity of the ashes was necessary to impart cleansing to the water.

Here is an excerpt from an ISBE Article giving more background on the practice of sprinkling in the OT...

Sprinkling (blood, water, oil) formed an important--if not the essential--part of the act of sacrifice. A consideration of the chief passages in the Old Testament will reveal the prominence and the significance of sprinkling as a feature of the sacrificial act. The significance of the sprinkling of blood is seen in the account of the establishment of the covenant between Yahweh and Israel (Ex 24:6-8). Half the blood was sprinkled on the altar as representing the Deity, while the remainder was put into a basin and then sprinkled on the people. This ceremony is a survival in a modified form of the communal meal in which the tribal god and his worshippers sat together and participated in the same food, and in this way came to possess the same life. The two-fold sprinkling of blood resulted in the establishment of an inviolable bond (Nu 18:17; 2 Ki 16:15).

In the account of the consecration of Aaron and his sons (Ex 29:16,20,21) the blood of the ram of the burnt offering was sprinkled on the altar, while the blood of the ram of consecration was put on the altar and sprinkled on Aaron and his sons and on their garments. Water of purifying was sprinkled on the Levites at their ordination (Nu 8:7).

Leviticus gives detailed information in regard to sacrificial sprinkling. In the case of burnt offering the blood was sprinkled round about upon the altar (Lev 1:5,11). The same practice obtained in the case of peace offerings, whether ox, lamb or goat (Lev 3:2,8,13). When a sin offering for sins inadvertently committed was made, the priest dipped his fingers in the blood and sprinkled it seven times before Yahweh, before the veil of the Holy Place (Lev 4:6). Elsewhere (Lev 16:11,15) we read that Aaron took the blood of the sin offering and sprinkled it with his finger upon the mercy-seat, eastward, 7 times (see also Nu 19:4).

Sprinkling constituted part of the process of purification. But it is obvious that the sprinkling, even in this case, was a religious act, and not part of the actual physical cleaning. A simple kind of sprinkler was made by fastening a bunch of hyssop to a cedar rod by a piece of scarlet thread or wool and then the patient was besprinkled 7 times (Lev 14:7), while oil was sprinkled with the finger, also 7 times, before Yahweh (Lev 14:16; see also Ex 12:22; Nu 19:18; Ps 51:7). The house in which the leper lived was disinfected in the same thorough manner (Lev 16:51).

In the case of persons who had contracted uncleanness through contact with a corpse, sprinkling with the "water of separation" was part of the process of cleansing. The water of separation consisted of the ashes of a red heifer (slain for the purpose) mixed with running water (Nu 19). A sprinkler was used as in the case of the leper (Nu 19:18). The final sprinkling--on the 7th day--was followed by a bath (Nu 19:19). The "tent" in which the corpse lay, together with all the contents, were thoroughly disinfected.

SANCTIFY FOR THE CLEANSING OF THE FLESH: hagiazei (3SPAI) pros ten tes sarkos katharothta: (Numbers 8:7; 19:12; 2 Chronicles 30:19; Psalms 51:7; Acts 15:9; 1 Peter 1:22)

Sanctify (37) (hagiazo from hagios = set apart from) means to set apart from profane things.

Cleansing (2514) (katharotes from katharos = pure) describes cleanness or purification. This ritual was a symbolic (and external) way of showing one was cleansed from sin.

Wuest writes that...

The unclean Israelite was, therefore, "out of bounds," so to speak, so far as participation in the tabernacle service of Israel was concerned, and also his service to God. When he fulfilled the Levitical ritual that had to do with his position and his restoration to a participation in the worship of Israel, he was sanctified, that is, set apart for God again.

In other words, he was cleaned up on the outside, but even that was only symbolic (not to mention imperfect and temporary). The writer's argument that is continued in verse 14 is that our Great High Priest in contrast to the ashes of a red heifer, effected internal cleansing (the conscience). Christ does much more than externally cleanse the old man for His blood paves the way for the birth of a brand new man. Paul wrote...

Therefore if any man is in Christ, he is a new creature; the old things passed away; behold, new things have come. (2Cor 5:17)

Peter speaks of the purification made possible by the blood of Christ...

Since you have in obedience to the truth purified your souls for a sincere love of the brethren, fervently love one another from the heart, 23 for you have been born again not of seed which is perishable but imperishable, that is, through the living and abiding word of God. (see note 1 Peter 1:22; 23)

There is a well known story of an evangelist named Alexander Wooten who related a story from the days when he held tent meetings. On one occasion after a series of meetings was over, he was pulling up his tent stakes when a young man approached him and asked what he had to do to be saved. The evangelist answered responded, “Sorry, it’s too late" which caused the young man to cry out "Oh no! Do you mean it’s too late because the services are over?” The evangelist explained that it was "too late" because it had already been done. In other words as he explained "Everything that could be done for your salvation has already been done.” After explaining Christ’s finished work to the young man, he led him to saving faith. Our salvation is based on the covenant whose redeeming work is finished-on a sacrifice that has been offered once and for all, that is complete and perfect and eternal. (Quoted by Warren Wiersbe in his comments on Dead John 19:28–30)

HOW MUCH MORE WILL THE BLOOD OF CHRIST WHO THROUGH THE ETERNAL SPIRIT OFFERED HIMSELF WITHOUT BLEMISH TO GOD: poso mallon to haima tou Christou hos dia pneumatos aioniou prosenegken (3SAAI) amomon to theo: (Deuteronomy 31:27; 2 Samuel 4:11; Job 15:16; Matthew 7:11; Luke 12:24,28; Romans 11:12,24) (12; 1 Peter 1:19; 1 John 1:7; Revelation 1:5)  (Isaiah 42:1; 61:1; Matthew 12:28; Luke 4:18; John 3:34; Acts 1:2; 10:38; Romans 1:4; 1 Peter 3:18) (Deuteronomy 33:27; Isaiah 57:15; Jeremiah 10:10; Romans 1:20; 1 Timothy 1:17) (Hebrews 9:7; 7:27; Matthew 20:28; Ephesians 2:5; 5:2; Titus 2:14; 1 Peter 2:24; 3:18) (Leviticus 22:20; Numbers 19:2-21; 28:3,9,11; Deuteronomy 15:21; 17:1; Isaiah 53:9; Daniel 9:24-26; 2 Corinthians 5:21; 1 Peter 1:19; 2:22; 1 John 3:5) (9; 1:3; 10:2,22)

Observe the three members of the Trinity in this passage -- Father, Son and Holy Spirit -- is involved. (Note: Although not everyone [eg, A T Robertson] agrees the "Eternal Spirit" is a reference to the Holy Spirit.)

How much more - Than the OT purification rituals which did in fact cleanse externally. The blood of Christ effects much more than this OT ritual for it results in internal cleansing of sin.

Robertson explains how much more writing that it is...

Instrumental case, "by how much more," by the measure of the superiority of Christ's blood to that of goats and bulls and the ashes of a heifer.

Wuest adds that...

The writer now makes a comparison between the efficacy of the blood of animals and that of the blood of Messiah. The former could cleanse ceremonial defilement, but the latter can cleanse from actual sin. And the reason why the blood of Messiah is so much more efficacious is stated by the writer in the words "Who through the eternal Spirit offered Himself without spot to God." (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Eerdmans or Logos)

Blood of Christ - Andrew Murray in The Holiest of All comments on the blood of Christ writing that...

The one element that gives the blood its value is, the holy obedience of which its outpouring was the proof; the blood of Christ who offered Himself without spot unto God. He came to live the life of man, such as God had meant Him to be, in creating Him. He gave up His will to God, He pleased not Himself but sought only God's pleasure, He yielded His whole life that God might reveal Himself in it as He pleased: He offered Himself unto God. He took and filled the place the creature was meant to fill. And that without spot. His self-sacrifice was complete and perfect, and His blood, even as the blood of a man, was, in God's sight, inexpressibly precious. It was the embodiment of a perfect obedience.

...It was the Word that became flesh, the Eternal Son of God who was made man. It was the life of God that dwelt in Him. That life gave His blood, each drop of it, an infinite value. The blood of a man is of more worth than that of a sheep. The blood of a king or a great general is counted of more value than hundreds of common soldiers. The blood of the Son of God!, it is in vain the mind seeks for some expression of its value; all we can say is, it is His own blood, the precious blood of the Son of God!

It was...(the) infinite worth of the blood that gave it such mighty power, first, in opening the grave, and then in opening heaven. It was this gave it the victory over all the powers of death and hell beneath, and gave Him the victor's place on high on the throne of God. And now, when that blood, from out of the heavenly sanctuary, is sprinkled on the conscience by the heavenly High Priest--how much more--with what an infinitely effectual cleansing, must not our conscience be cleansed. (The Holiest of All) (Bolding added)

Spurgeon in his sermon (1 Peter 1:19 The Precious Blood of Christ) on the precious blood of Christ writes that

in Hebrews 9:14 Christ sanctified the people by His own blood. Certain it is, that the same blood which justifies by taking away sin, does in its after-action act upon the new nature and lead it onward to subdue sin and to follow out the commands of God. There is no motive for holiness so great as that which streams from the veins of Jesus. If you want to know why you should be obedient to God's will, my brethren, go and look upon Him who sweat, as it were, great drops of blood, and the love of Christ will constrain you, because you will thus judge, "That if one died for all, then were all dead: and that he died for all, that we which live might not henceforth live unto ourselves, but unto him that died for us and rose again."

Spurgeon (Hebrews 9:19-20 The Blood of the Testament -3293) writes...

I do not think anyone ever knows the preciousness of the blood of Christ, till he has had a full sight and sense of his sin, his uncleanness, and his ill-desert. Is there, any such thing as truly coming to the cross of Christ until you first of all have seen what your sin really deserves? A little light into that dark cellar, sir; a little light into that hole within the soul, a little light cast into that infernal den of your humanity, and you will soon discern what sin is, and, seeing it, you would discover that there was no hope of being washed from it except by a sacrifice far greater than you could ever render. Then the atonement of Christ would become fair and lustrous in your eyes, and you would rejoice with joy unspeakable in that boundless love which led the Savior to give himself a ransom for us, "the Just for the unjust, that he might bring us to God." May the Lord teach us, thundering at us, if need be, what sin means. May he teach it to us so that the lesson shall be burned into our souls, and we shall never forget it! I could fain wish that you were all burden-carriers till you grew weary. I could fain wish that you all laboured after eternal life until your strength failed, and that you might then rejoice in him who has finished the work, and who promises to be to you All-in-all when you believe in him, and trust in him with your whole heart.

Eternal (166) (aionios from aion) means perpetual eternal, everlasting, without beginning or end (as of God), that which is always. Eternal is a key word Hebrews: blood of eternal covenant (see note Hebrews 13:20). He offered Himself through His eternal spirit (see note Hebrews 9:14) and has become the Author/Source of eternal salvation (see note Hebrews 5:9). He has obtained eternal redemption (see note Hebrews 9:12) and enables men to receive of the eternal inheritance (see notes Hebrews 9:15; 13:20).

John MacArthur writes that

Jesus did everything He did on earth in obedience to the Father through the Spirit. Even in His supreme sacrifice He through the eternal Spirit offered Himself without blemish to God. In doing so, He provided the cleansing of our consciences from dead works to serve the living God. He frees our consciences from guilt, a joy and a blessing that no Old Testament saint ever had or could have had. In Christ we can “draw near with a sincere heart in full assurance of faith, having our hearts sprinkled clean from an evil conscience and our bodies washed with pure water” (Heb 10:22). (MacArthur, John: Hebrews. Moody Press or Logos)

Cole writes that...

Scholars debate whether “eternal Spirit” refers to the Holy Spirit or to Jesus’ eternal divine spirit (there were no capital letters in the original Greek). We cannot be dogmatic on this. If it refers to the Holy Spirit, then it means that Jesus relied on the Holy Spirit when He went to the cross, which is certainly true. If it refers to Jesus’ eternal divine nature, the emphasis would be on the fact that Jesus’ sacrifice was uniquely efficacious to redeem His people, be-cause He is not only a man, but also is eternal God (7:3, 16). The point is, “the difference between the Levitical offerings and Christ’s self-offering was infinite rather than relative” (P. Hughes, p. 360). This infinitely efficacious sacrifice satisfied God in a way that the blood of bulls and goats never could. Through Christ’s blood, we can have a clean conscience.  (Hebrews 9:1-14 God's Remedy for Guilt)

Offered Himself - This emphasizes once again the voluntary character of Christ's death. He was not coerced but gave Himself willingly for us because He loved us and did not want us to suffer eternal separation from the Father.

Jeffrey Ebert has the following illustration...

When I was 5 years old, before factory-installed seat belts and automobile air bags, my family was driving home at night on a two-lane country road. I was sitting on my mother's lap when another car, driven by a drunk driver, swerved into our lane and hit us head-on. I don't have any memory of the collision. I do recall the fear and confusion I felt as I saw myself literally covered with blood from head to toe.  Then I learned that the blood wasn't mine at all, but my mother's. In that split second when the two headlights glared into her eyes, she instinctively pulled me closer to her chest and curled her body around mine. It was her body that slammed against the dashboard, her head that shattered the windshield. She took the impact of the collision so that I wouldn't have to. It took extensive surgery for my mother to recover from her injuries.  In a similar, but infinitely more significant way, Jesus Christ took the impact for our sin, and his blood now permanently covers our lives. (Fresh Illustrations for Preaching & Teaching. Baker)

CLEANSE YOUR CONSCIENCE FROM DEAD WORKS: kathariei (3SFAI) ten suneidesin hemon apo nekron ergon: (Hebrews 9:9, 9:14, 10:22, 13:18) (see note Hebrews 6:1)

Cleanse (2511) (katharizo from katharos = pure, clean, without stain or spot; English words - catharsis = emotional or physical purging, cathartic = substance used to induce a purging, Cathar = member of a medieval sect which sought the purging of evil from its members) means to make clean by taking away an undesirable part. To cleanse from filth or impurity. Click here (and here) for more background on the important Biblical concept of clean and cleansing.

In secular Greek katharizo occurs in inscriptions for ceremonial cleansing.  This word group conveys the idea of physical, religious, and moral cleanness or purity in such senses as clean, free from stains or shame, and free from adulteration.

And behold, a leper (see Leviticus 13) came to Him (Jesus), and bowed down to Him, saying, "Lord, if You are willing, You can make me clean (katharizo)." And He stretched out His hand and touched him, saying, "I am willing; be cleansed (katharizo - aorist imperative)" And immediately his leprosy was cleansed (katharizo). (Matthew 8:2-3)