Certainly this must not be taken in a blasphemous manner to imply that Christ was, as Martin Luther claimed, a sinner, or even the worst of sinners, one who must have experienced and committed murder, adultery, sacrilege, and all the depths of depravity that mankind has cast ourselves into. There may be a certain psychological cogency to such an argument, but Christ nonetheless is not Baudelaire, and the spiritual illumination sought by Rimbaud on the left-hand path of the "systematic derangement of the senses" did not bear a redemptive nature, much less a nature that could redeem the sins of the world. Christ was absolutely sinless, as was the All-Holy One, the Panagia, His mother. In an effort to ground her (and Him) fully in the human experience, many Orthodox will turn to the homily of St. Gregory Palamas on the Forefathers, and see in that homily a witness to the "progressive gracing" of the ancestors of Our Lord, of the Incarnation as the pinnacle and culmination of human history.
Nonetheless, the question remains, "how could Our Lord have become incarnate in a fallen human nature? What does this mean?"
It does not mean that Jesus Christ ever committed sin.
It does not mean that Jesus Christ was ever tempted by sin. Certainly He was tempted by sin, not only in the prototypal Lenten fast in the desert but also in the Garden of Gethsemane. Yet Adam and Eve were also tempted by sin, and they were not fallen. Therefore, temptation is not what is entailed by fallenness, but rather something incidental to it.
It does not mean that the bodies of either Our Lord or the Panagia suffered corruption in the grave. The Byzantine liturgical texts on both feasts witness abundantly that they did not.
It does not mean inordinate derangement of the spiritual faculties or subjection to the passions. Both the Theotokos and her Son were tempted, and the Theotokos, as a creature not yet fully divinized in what the Latin West will call the "Beatific Vision", possessed gnomic will (the faculty of choosing among apparent goods with the concomitant ability to sin, discussed by St. Maximos the Confessor). Yet both also possessed supreme tranquility of soul and the peace which surpasses all understanding.
It does mean death. The death of the Theotokos was necessary, as was the death of Christ. The necessity of their deaths in no way impinges upon their free choice to die (as argued from St. Dimitri of Rostov, here: http://byzantinechesterton.blogspot.com/2011/08/necessity-of-dormition-of-theotokos.html). The death of the all-holy Theotokos was peaceful, painless and sinless. And yet, the death of Christ was not.
If death is what fundamentally the fallenness of their human natures entailed, then the fallenness of the human nature of Christ is something essentially marked by the Passion and Redemption. The fallenness of the human nature of Christ, therefore, is something by nature sacrificial.
Blessed Columba Marmion says the following words in Christ the Ideal of the Priest, page 22 (Glasgow: Sands & Co., 1952):
So, coming into the world, the Son of God assumed a "sacrificial body" suited for enduring suffering and death. He was truly a member of the human race, like us, and it is in the name of His brethren that He is to offer Himself as victim to reconcile them with their Father in heaven.