Whenever you observe a term of conclusion like "for this reason", "because", "for", "therefore", etc, always take a moment and ask the natural question "for what reason?" You will usually be led to read the preceding verses (the immediate context). What attitude/action is the author calling for in his reader's in view of this truth ("this reason")?

So first, what is "this reason" (if you have time make your own observations of Hebrews 1 before you read the following notes)? Jesus is superior to prophets and angels and because of this superior revelation, the reader must make an appropriate response. As the writer of Hebrews will do throughout this epistle, after presenting a doctrinal truth, he makes a direct application of that truth to the experience of his readers. This passage presents the first of those many applications.

A W Pink adds that...

Therefore, signifies, for this cause: because God has vouchsafed (granted or furnish often in a gracious or condescending manner) so excellent a Teacher (His Son), He must be the more carefully attended unto. The "therefore" looks back to all the varied glories which set forth Christ’s excellency named in the previous chapter.

Because He is God’s "Son," therefore give heed.
Because He is "the Heir of all things," therefore give heed.
Because He "made the worlds," therefore give heed; and so on.

These are so many grounds on which our present exhortation is based. “Therefore is equivalent to, ‘Since Jesus Christ is as much better than the angels, as He both received by inheritance a more excellent name than they—since He is both essentially and officially inconceivably superior to these heavenly messengers, His message has paramount claims on our attention, belief, and obedience’,” (Dr. J. Brown). (Hebrews 2:1-4 Christ Superior to Angels)

It was usual with the prophets to preface their utterances with a “Thus saith the Lord,” and thereby arrest the attention and awe the hearts of their hearers. Here the writer refers to the Person of the Lord Himself as the argument for hearing what He said. In short, the new revelation in Christ is superior to the old. You have received a revelation superior to that given in the old testament, and it has been given to you through One Who is superior to the angels. Now you must respond to this truth!

It is noteworthy that Hebrews 2:1-4 forms a "parenthesis", interrupting the discussion of Christ’s relationship to angels, which is resumed in Hebrews 2:5.

A T Robertson adds

Because Jesus is superior to prophets and angels and because the new revelation is superior to the old. The author often pauses in his argument, as here, to drive home a pungent exhortation.

The author often pauses in his argument, as here, to drive home a pungent exhortation. The revelation spoken in the Son is the final and complete revelation, given through Christ Who is superior to prophets and angels.

The author gives a strong exhortation to his readers to respond to what they have just read in Hebrews 1. In a sense we see sound doctrine flow seamlessly into serious invitation. If you are a teacher, remember that to be effective, you must do more than present biblical facts. Using sound doctrine as your foundation, you must also warn, exhort, invite. And so we note that by the time the writer of Hebrews gets to chapter 2 he is impassioned and concerned the salvation of his hearers. Thus he exhorts his readers to respond to what he says. In short, as any good Biblical teacher, he not only seeks for Christ to be seen as superior but also to be accepted as Savior. His invitation includes both exhortation and warning, what to do and what happens if you don't respond.

In the last chapter of Hebrews the writer sums up his message, writing

I urge you, brethren, bear with this word of exhortation, for I have written to you briefly. (see note Hebrews 13:22)

God’s Word demands a response, and a faithful teacher of the Word teaches for a response.

Matthew Henry explains the writer's method this way:

The apostle proceeds in the plain profitable method of doctrine, reason, and use, through this epistle. Here we have the application of the truths before asserted and proved; this is brought in by the illative particle therefore, with which this chapter begins, and which shows its connection with the former, where the apostle having proved Christ to be superior to the angels by whose ministry the law was given, and therefore that the gospel dispensation must be more excellent than the legal, he now comes to apply this doctrine both by way of exhortation and argument.

WE MUST PAY MUCH CLOSER ATTENTION: dei (3SPAI) perissoteros prosechein (PAN): (Deuteronomy 4:9,23; 32:46,47; Joshua 23:11,12; 1 Chronicles 22:13; Psalms 119:9; Proverbs 2:1-6; Proverbs 3:21; 4:1-4,20-22; 7:1,2; Luke 8:15; 9:44)

Thus we are bound to pay all the more heed to what we have been told, for fear of drifting from our course. (New English Bible)

Must (1163) (dei from deo= to bind, tie objects together) means this is necessary (binding) or needful. Dei marks a logical necessity and not a moral obligation: we must rather than we ought. It therefore speaks of an obligation out of intrinsic necessity or inevitability. It is necessary that this happen. It is continually (present tense) necessary...to pay close attention.

The verb must is very strong and emphatic and calls for the reader to take responsibility, giving us an exhortation to perform a specific duty.

Why? So we do not drift. The verb is in the active voice meaning that we must decide to do it. It is a choice we must make. To fail to do so can have eternally damning consequences!

When the Son of God speaks to people, every consideration makes it appropriate that we should attend to what is spoken. Truth demands a response: “Now what are you going to do about this?” We can know all the truth there is to know about Jesus Christ and yet go to hell if we never make Him our own by being made His own.

The writer of Hebrews uses this strong verb must (dei) again in his explanation of how we can "please" God writing that...

"without faith it is impossible to please Him, for he who comes to God must (dei) believe that He is, and that He is a rewarder of those who seek Him." (see note Hebrews 11:6)

We must be sure to understand that it is not the gospel that slips, as the King James Version seems to imply (It reads "Therefore we ought to give the more earnest heed to the things which we have heard, lest at any time we should let them slip."). That is not the meaning at all. The Greek and most modern translations make it clear that it is inattentive men who slip. The Word will never drift from us.

Pay...attention (4337) (prosecho from pros =  toward + echo = to have, hold) means literally "to hold toward", to hold (the mind) to, to apply one’s self to, to attach one’s self to. The idea is to "take hold" of something and pay attention to it, "putting your mind on it", especially in the sense of being on guard. It means to turn the mind to something and includes the idea of to act upon what one perceives (cf. Acts 8:6; 16:14).

Prosecho means to be in a continuous state of readiness to learn of any future danger, need, or error, and to respond appropriately - keep on the lookout for, be alert for, be on one's guard against. Prosecho almost always warns of danger. Prosecho is not a call simply to notice or sense something, but to be on guard against it because it is harmful.

To pay...attention is to apply the mind to a particular subject, to attend to it, to consider it and in the present use is opposite of neglect of a so great salvation.

The Preacher's Commentary adds that...

Our author’s antidote for such “drifting” or slipping away consists of earnest heed, intense concentration, thoughtful clarification. By such means the roots of an idea get deep into our consciousness and consciences. (Briscoe, D. S., & Ogilvie, L. J. The Preacher's Commentary Series,  New Testament. 2003; Thomas Nelson or Logos)

A W Pink in quoting Dr Gouge explains prosecho writing that...

The duty here intended is a serious, firm, and fixed settling of the mind upon that which we hear; a bowing and bending of the will to yield unto it; an applying of the heart to it, a placing of the affections upon it, and bringing the whole man into conformity thereunto. Thus it comprises knowledge of the Word, faith therein, obedience thereto, and all other due respects that may any way concern it.

Prosecho was commonly used to describe the bringing of a ship to land. What a striking picture this secular use congers up - Think of our life as a "ship of faith" moving through the fog, avoiding the dangerous reefs by keeping our focus on the Lighthouse on the shore. Clinging with a devotion to the truth about Jesus. Holding the rudder firmly in place towards the safe harbor of the Light of the World.

It is interesting that the two main verbs in this verse (prosecho and pararrhueo) have nautical uses. Prosecho as mentioned above can mean to moor a ship whereas pararrhueo was used to describe a ship that had been carelessly allowed to slip past a harbour or a haven because the mariner has forgotten to allow for the wind or the current or the tide! This verse could be paraphrased

Therefore, we must the more eagerly anchor our lives to the things that we have been taught lest the ship of life drift past the harbour of salvation and be wrecked and lost forever.

What a picture...the ship slowly drifts to destruction because the pilot is asleep at the wheel! How apropos this picture is in regard to how most unbelievers simply drift along "the way (that) is broad (and) that leads to destruction". Many are the drifters who eventually plunge into everlasting torment and separation from God. The point is that the majority of those who enter by the "wide gate" do not go headlong, wantonly and intentionally into hell. Most people who populate hell haven't been the most profane God haters and vile  rejecters of His Son. Most of those who are deceived and dead in their trespasses and sins just slowly, almost imperceptibly slip past the harbor of salvation and into the "sea" of eternal destruction. One writer phrases it this way...

There is a tide in the affairs of men which, taken at its ebb, leads to victory; neglected, the shores of time are strewn with the wreckage.

The writers point is that his reader (which includes all of us) must put our mind to the Son of God's teaching on salvation lest we too one day find ourselves having forever drifted past the harbor of salvation. Salvation once gained cannot be lost. The writer's point is that the reader must be certain that genuine salvation has been gained!

Guzik comments that...

Give the more earnest heed has not only the idea of hearing carefully, but also in doing what we have heard - and we must give the more earnest heed. (Hebrews 2)

Peter uses prosecho with a similar thought explaining that

"we have the prophetic word made more sure, to which you do well to (present tense - continually) pay attention as to a lamp shining in a dark place, until the day dawns and the morning star arises in your hearts." (see note 2 Peter 1:19)

Much close" (4057) (perissoteros) is a comparative adverb which can be translated "all the more", "even much more", "more earnestly," "more superabundantly". And so one could render it "we must all the more pay close attention".

The College Press NIV commentary has an interesting note writing that while

"The word “better” could be written across the book of Hebrews; the word "more" should be written across our response. Some things require greater (perissoteros, “far more, to a much greater degree”) diligence." (Girdwood, J., & Verkruyse, P. The College Press NIV commentary Joplin, Mo: College Press)

I like William Barclay's rendition of this section:

We must, therefore, with very special intensity pay attention to the things that we have heard.

Why? God has spoken in His Son. We must continually hold to the Words of Truth spoken by the Son Who alone is Truth. There is nothing else to that needs to be said! No more revelation is forthcoming for none is necessary.

TO WHAT WE HAVE HEARD: hemas tois akoustheisin (APPNPD):

We have heard (191) (akouo) means not just to hear sounds per se but implies hearing with attention or hearing so to speak with the "ear of one's mind". “To hear” implies “to obey.” This verb is more literally translated in this verse as

"to what we have been hearing"

As Pink notes that simply...

To “hear” is not sufficient. There must be prayerful meditation, personal appropriation.

He uses this verb akouo in the next verse writing that "it was confirmed to us by those who heard" which implies that they heard effectively or else they could not have passed it on.

In Hebrews 1 we have just heard the truth about Jesus, as summarized below

Creator
Heir of all
One with the Father yet distinct in Person
Upholder of all things by His Word
Purifier from sin
Seated at the Right hand of the Majesty on high
Having a better name than the angels (begotten the Son of God)
God testifying He is His Father fulfilling the Davidic Covenant
Firstborn = His preeminence,
God, Lord, King, Victorious Warrior
Immutable
Eternal
Better than angels who serve Him, worship Him, are created by Him & are sent out to serve believers.

Having heard, have you responded? If not, will you respond in faith right now? Or will you be like those the writer records in chapter 4 warning that...

indeed we have had good news preached to us, just as they also; but the word they heard (akouo) did not profit them, because it was not united by faith in those who heard. (Hebrews 4:2)

Hearing does not necessarily equate with believing. Jesus declared:

Truly, truly (Amen, Amen = trustworthy, trustworthy), I say to you, he who hears  (akouo) My word, and believes Him who sent Me, has eternal life, and does not come into judgment, but has passed out of death into life. (John 5:24)

Dear reader don't drift past such a great Savior Who offers a great salvation.

The idea of hearing is a key idea in Hebrews...

Hebrews 2:1 (note) For this reason we must pay much closer attention to what we have heard, lest we drift away from it.

Hebrews 3:7 (note) Therefore, just as the Holy Spirit says, "TODAY IF YOU HEAR HIS VOICE, 8 DO NOT HARDEN YOUR HEARTS AS WHEN THEY PROVOKED ME, AS IN THE DAY OF TRIAL IN THE WILDERNESS

Hebrews 3:15 (note) while it is said, "TODAY IF YOU HEAR HIS VOICE, DO NOT HARDEN YOUR HEARTS, AS WHEN THEY PROVOKED ME."

Hebrews 4:7 (note) He again fixes a certain day, "Today," saying through David after so long a time just as has been said before, "TODAY IF YOU HEAR HIS VOICE, DO NOT HARDEN YOUR HEARTS."

Hebrews 5:9 (note) And having been made perfect, He became to all those who obey (literally "hear under",  listen attentively hupakouo = hupo + akouo) Him the source of eternal salvation,

Hebrews 5:11 (note) Concerning him we have much to say, and it is hard to explain, since you have become dull of hearing.

Hebrews 11:8 (note) By faith Abraham, when he was called, obeyed (literally "hear under",  listen attentively hupakouo = hupo + akouo) by going out to a place which he was to receive for an inheritance; and he went out, not knowing where he was going.

LEST WE DRIFT AWAY: mepote pararhuomen (1PAPS): (Heb 12:5; Matthew 16:9; Mark 8:18; 2 Peter 1:12,13,15; 3:1) (Habakkuk 1:6; 2:16)

Most authors agree that this first of five major warning passages alerts the readers to the danger of apostasy, a theme that is echoed in all five warning passages.

Hebrews 2:1-3
Hebrews 3:1-4:16
Hebrews 5:11-6:20
Hebrews 10:19-39
Hebrews 12:15-29

Bob Smith in well written online text Basics of Bible Interpretation uses the warning passages of Hebrews to illustrate the importance of considering the cultural/historical context as a guide to accurate interpretation...

We are prone to interpret everything we read in terms of our twentieth century Western culture, since that's the sphere in which we live. It takes a conscious effort to research and absorb some of the data that will make our thoughts conform to the time and culture of the writer. Much of this can be accomplished through the use of Bible Dictionaries, and books on the history and archeology of Old and New Testament times. Edersheim's The Life and Times of Jesus the Messiah (1883) is a classic work on the cultural features of biblical times. Old and New Testament introductions also help us get the feel and flavor of life in Bible times. The bibliography in the Appendix will give you further source materials on this subject.

To illustrate the principle we have before us: it is helpful, when studying the Book of Hebrews, to remember that it was written while the temple was still standing in Jerusalem. This makes clear the need for the strong appeal to these early Hebrew Christians to "go forth to him [Jesus] outside the camp, bearing abuse for him. For here we have no lasting city, but we seek the city which is to come" (see notes Hebrews 13:13; 13:14).

And, "We have an altar from which those who serve the tent have no right to eat" (see note Hebrews 13:10).

There was strong temptation for these early Christians to be drawn back into the "shadows" of the temple worship forms and away from the reality of Christian faith.

Observing this, we can better understand the strong warnings in the book. After all, it was written to the Hebrews of the first century. Our interpretation should be based on this historical/cultural setting, with application to our contemporary Western scene flowing out of that interpretation. It then becomes obvious that there are many similar situations where modern men and women halt short of true faith, settling for religious ritual instead. The same strong warnings can then be applied properly to present situations. The following outline illustrates how applicable this truth is, almost twenty centuries after its writing.

DON'T MISS MELCHIZEDEK!

We need to advance from the Aaronic priesthood (the Law) to the Melchizedek priesthood, which portrays the adequacy and resources of Christ as our risen, living Lord. In view of Christ's availability to us as our great High Priest after the order of Melchizedek, to strengthen us for every test and supply our every need, consider these warnings from Hebrews:

1. THE DANGER OF DRIFTING, Hebrews 2:1-3

The Problem: Inattention
The Possibility: Of drifting.
The Test: Are You Listening?

2. THE DANGER OF AN UNBELIEVING HEART, Hebrews 3:12-4:16

The Problem: Hearing, but not Believing
The Possibility: Hardening of the heart.
The Test: How Well Do You Rest?

3. THE DANGER OF PROLONGED IMMATURITY, Hebrews 5:11-6:20

The Problem: Being Long-Time Babies
The Possibility: Missing the value of Melchizedek's priesthood.
The Test: Are You Really Going Anywhere?

4. THE DANGER OF MISSING REALITY--Seeing just the shadow, Hebrews 10:26-31

The Problem: Failing to Respond to Grace
The Possibility: of spurning the Son.
The Test: How Real Is Jesus Christ to You?

5. THE DANGER OF CONTRADICTION, Hebrews 12:15-29

The Problem: The Practical Denial of Christ's Lordship
The Possibility: Of refusing Him the right to give orders.
The Test: How Well Do You Worship through Obedience to Christ?

(From the online text - Bob Smith's well written Basics of Bible Interpretation)

Kent Hughes is correct when he says that the...

church's experience 2,000 years ago intersects our lives in this way: drifting is the besetting sin of our day. And as the metaphor suggests, it is not so much intentional as from unconcern. Christians neglect their anchor-Christ-and begin to quietly drift away. There is no friction, no dramatic sense of departure. But when the winds of trouble come, the things of Christ are left far behind, even out of sight. (Hughes, R. K. Hebrews: An Anchor for the Soul. Volume 1.  Crossway Books; Volume 2 or Logos)

See more quotations on - Backsliding or Drifting

Spurgeon wrote it is as if the writer had said...

Seeing Christ is so excellent in His person, and seeing the Gospel has such a glorious Author, let us take great care that we esteem His person, revere His authority, reverence His ministry, and believe His message; and let us take heed that our memories be not like leaking vessels, suffering the word at any time to slip or run from us.

We have heard them; do not let us forget them. Let them not be like the driftwood which goes floating down the stream. Let us make a desperate effort to retain them in our memories; and, above all, to ponder them in our hearts.

Drift away (3901) (pararrhueo from pará = by, past, beyond + rhéo = flow) literally means to "flow past" and so to glide, to be washed away, to drift away. Figuratively as in the present verse pararrhueo  means to slip or drift away from belief. It is a picture of a gradual, almost unnoticed movement past a certain point. It describes that carelessness of mind which, perhaps occupied by other things, is not aware it is losing ground. It is like a ship drifting without anchorage and so drifting away from its moorings and from a place of safety to one of danger.

Pararrhueo  is used only here in the NT and twice in the Septuagint (LXX)...

Proverbs 3:21 My son, let them (in the immediate context, "them" is God's wisdom, understanding and knowledge!) not depart from your sight; Keep sound wisdom and discretion

Isaiah 44:4 And they will spring up among the grass Like poplars by streams (pararrhueo  - running or flowing) of water.'

Pararrhueo  was used to describe mooring of ships (to moor a ship means to make it secure with cables, lines, or anchors) and the drift caused by winds and tides if ships are not moored.

Kent Hughes writes that...

I have experienced this firsthand while fishing the tidal inlets of the California coast, when winds or surging tides have imperceptibly slipped the anchor from the seabed so that it hung suspended, and I, intent on my fishing, unknowingly moved several hundred yards and almost foundered on the rocks! Such dangerous drifting is not intentional, but comes rather from inattention and carelessness—which was precisely the problem with the pressured little church. They had become careless about their moorings in Christ. At first, in calm waters, that was not noticeable. But as the storms of opposition rose, some of them were drifting farther and farther away from Christ toward the shoals of shipwreck in their old world of Judaism." (Hughes, R. K. Hebrews: An Anchor for the Soul. Volume 1.  Crossway Books & Volume 2)

The Greek writer Xenophon used pararrhueo to describe the river flowing by a certain place. It was used to picture the snow slipping off from the soldiers' bodies, of a ring slipping from one's finger or of a thought slipping out of one's memory. This verb thus presents a vivid picture of individuals who let themselves drift away from the haven of the gospel of Christ. One need not be violently opposed to the message to suffer loss but just to "drift" away from it!

Ray Stedman writes that...

The danger highlighted is that of a great loss occurring unnoticed. The cause is not taking seriously the words spoken to them. Inattention or apathy will rob them of their treasure. With these words, the writer reveals his shepherd’s heart, since he is not content with instructing the mind with intriguing doctrine. He also longs to reach the heart and move the will to action...It is not necessary to openly renounce the gospel. One can remain lost by simply and quietly drifting away from hearing it, or hearing it with no comprehension of the seriousness of its message." (