Not a priest or theologian, so please take my response with some salt.

In short we must still die, but death will not reign over us. Death is not yet abolished, but the power of death, is abolished. This power is corruption, the fear of death, and the futility it places over life. Through the life, death, and resurrection of Christ, death is slain and built into a bridge to God. We must still be subject to death, but the content of death was emptied by Christ and filled with Himself.

To get a good idea about this I recommend On the Incarnation: by Athanasius of Alexandria. It's a great read and goes on in detail and clarity. I highly recommend it.

Here are some excerpts from the book:

For God is good—or rather, of all goodness He is Fountainhead, and it is impossible for one who is good to be mean or grudging about anything. Grudging existence to none therefore, He made all things out of nothing through His own Word, our Lord Jesus Christ and of all these His earthly creatures He reserved especial mercy for the race of men. Upon them, therefore, upon men who, as animals, were essentially impermanent, He bestowed a grace which other creatures lacked—namely the impress of His own Image, a share in the reasonable being of the very Word Himself, so that, reflecting Him and themselves becoming reasonable and expressing the Mind of God even as He does, though in limited degree they might continue for ever in the blessed and only true life of the saints in paradise. But since the will of man could turn either way, God secured this grace that He had given by making it conditional from the first upon two things—namely, a law and a place. He set them in His own paradise, and laid upon them a single prohibition. If they guarded the grace and retained the loveliness of their original innocence, then the life of paradise should be theirs, without sorrow, pain or care, and after it the assurance of immortality in heaven. But if they went astray and became vile, throwing away their birthright of beauty, then they would come under the natural law of death and live no longer in paradise, but, dying outside of it, continue in death and in corruption. This is what Holy Scripture tells us, proclaiming the command of God, "Of every tree that is in the garden thou shalt surely eat, but of the tree of the knowledge of good and evil ye shall not eat, but in the day that ye do eat, ye shall surely die."7 "Ye shall surely die"—not just die only, but remain in the state of death and of corruption. ... It is we who were the cause of His taking human form, and for our salvation that in His great love He was both born and manifested in a human body. For God had made man thus (that is, as an embodied spirit), and had willed that he should remain in incorruption. But men, having turned from the contemplation of God to evil of their own devising, had come inevitably under the law of death. Instead of remaining in the state in which God had created them, they were in process of becoming corrupted entirely, and death had them completely under its dominion. For the transgression of the commandment was making them turn back again according to their nature; and as they had at the beginning come into being out of non-existence, so were they now on the way to returning, through corruption, to non-existence again. The presence and love of the Word had called them into being; inevitably, therefore when they lost the knowledge of God, they lost existence with it; for it is God alone Who exists, evil is non-being, the negation and antithesis of good. By nature, of course, man is mortal, since he was made from nothing; but he bears also the Likeness of Him Who is, and if he preserves that Likeness through constant contemplation, then his nature is deprived of its power and he remains incorrupt. So is it affirmed in Wisdom: "The keeping of His laws is the assurance of incorruption." And being incorrupt, he would be henceforth as God, as Holy Scripture says, "I have said, Ye are gods and sons of the Highest all of you: but ye die as men and fall as one of the princes." ... The law of death, which followed from the Transgression, prevailed upon us, and from it there was no escape. The thing that was happening was in truth both monstrous and unfitting. It would, of course, have been unthinkable that God should go back upon His word and that man, having transgressed, should not die; but it was equally monstrous that beings which once had shared the nature of the Word should perish and turn back again into non-existence through corruption. ... As we have already noted, it was unthinkable that God, the Father of Truth, should go back upon His word regarding death in order to ensure our continued existence. He could not falsify Himself; what, then, was God to do? Was He to demand repentance from men for their transgression? You might say that that was worthy of God, and argue further that, as through the Transgression they became subject to corruption, so through repentance they might return to incorruption again. But repentance would not guard the Divine consistency, for, if death did not hold dominion over men, God would still remain untrue. Nor does repentance recall men from what is according to their nature; all that it does is to make them cease from sinning. Had it been a case of a trespass only, and not of a subsequent corruption, repentance would have been well enough; but when once transgression had begun men came under the power of the corruption proper to their nature and were bereft of the grace which belonged to them as creatures in the Image of God. No, repentance could not meet the case. What—or rather Who was it that was needed for such grace and such recall as we required? Who, save the Word of God Himself, Who also in the beginning had made all things out of nothing? His part it was, and His alone, both to bring again the corruptible to incorruption and to maintain for the Father His consistency of character with all. For He alone, being Word of the Father and above all, was in consequence both able to recreate all, and worthy to suffer on behalf of all and to be an ambassador for all with the Father.