For since they were Jews and learned from the fathers that they must expect both their good and their evil immediately and must live accordingly, but then [when the Gospel came] the opposite was [taught]--their good things being in hope and after death, their evils in hand, though they had patiently endured much, it was likely that many would be fainthearted;--hereon he discourses.
But we will unfold these things at a fit opportunity. At present: he of necessity wrote to those for whom he cared so greatly. For while the reason why he was not sent to them is plain, yet he was not forbidden to write. And that they were becoming fainthearted he shows when he says, "Lift up the hands which hang down, and the feeble knees, and make straight paths" and again, "God is not unrighteous to forget your work and love." For the soul overtaken by many trials, was turned aside even from the faith. Therefore he exhorts them to "Give heed to the things which they have heard, and that there should not be an evil heart of unbelief." On this account also, in this Epistle, especially, he argues at length concerning faith, and after much [reasoning] shows at the end that to them [of old] also He promised good things in hand, and yet gave nothing.
And besides these things, he establishes two points that they might not think themselves forsaken: the one, that they should bear nobly whatever befalls them; the other, that they should look assuredly for their recompense. For truly He will not overlook those with Abel and the line of unrewarded righteous following him.
And he draws comfort in three ways: first, from the things which Christ suffered: as He Himself says, "The servant is not greater than his Lord." Next, from the good things laid up for the believers. Thirdly, from the evils; and this point he enforces not only from the things to come (which would be less persuasive), but also from the past and from what had befallen their fathers. Christ also does the same, at one time saying, "The servant is not greater than his Lord"; and again, "There are many mansions with the Father"; and He denounces innumerable woes on the unbelievers.
But he speaks much of both the New and the Old Covenant; for this was useful to him for the proof of the Resurrection. Lest they should disbelieve that [Christ] rose on account of the things which He suffered, he confirms it from the Prophets, and shows that not the Jewish, but ours are the sacred [institutions]. For the temple yet stood and the sacrificial rites; therefore he says, "Let us go forth therefore without, bearing His reproach." But this also was made an argument against him: "If these things are a shadow, if these things are an image, how is it that they have not passed away or given place when the truth was manifested, but these things still flourish?" This also he quietly intimates shall happen, and that at a time close at hand.
Moreover, he makes it plain that they had been a long time in the faith and in afflictions, saying, "When for the time ye ought to be teachers," and, "Lest there be in any of you an evil heart of unbelief," and ye became "Followers of them who through patience inherit the promises."
Source: Homilies on the Epistle to the Hebrews (New Advent)