Atonement--it's a word that typically brings to mind images of sacrifice, of offering up an animal as payment for sin. But, this (which we call substitutionary atonement) is really only one aspect of atonement. Buried within the commandments for the Day of Atonement, hidden away in Jewish tradition, alluded to by Messiah, and expanded upon by Paul we find another aspect of atonement which calls upon us as participants--vicarious atonement.
Most of Leviticus 16 is devoted to describing the Temple service and its associated sacrifices, but at the end is a small section describing the responsibilities of the children of Israel. There we are told to "afflict yourselves." This should bring to mind Messiah's words, "If any man will come after me, let him deny himself, take up his cross, and follow me" (Matt. 16:24). If we want to participate in the atonement sacrifice, we must humble ourselves, becoming as dust before our King.
Why? It almost sounds like some sort of transactional reward system--if you do this, you get that. But that is not the case at all. That would be a system of works. No, the denial of self is not a way of earning atonement, but rather is an integral part of the atoning process itself. To understand this, let's look at the sacrifices through a traditional lens.
Chasidic Judaism paints a sacrificial picture which is vicarious--i.e. we act through the sacrifice and experience the result ourselves. When we slay the animal and place it on the altar, we are to understand that we are vicariously placing ourselves on that altar. We are investing our own identity into the animal and through that process putting ourselves to death. In some explanations, it is our "animal soul" (or as Paul would put it, our "flesh") specifically that we are putting upon that alter. So, when we deny ourselves on Yom Kippur, we can understand that we are in a sense dying through the atonement sacrifice.
Paul makes this explicit in Galatians 2:20, when he says "I have been crucified with Christ." Yes, Messiah died in our stead, paying the penalty that we could never repay, but also in doing so, paved a path through which we might be united with Him in death. Still, the story does not end there. This continues toward a purpose, which Paul explains in Romans 6, "We were buried therefore with him by baptism into death, in order that, just as Christ was raised from the dead by the glory of the Father, we too might walk in newness of life."
This is where it is all pointing. Resurrection. Restoration. Redemption. The old self is crucified that the new self might be made alive. We are a new creation. In this, death and sin are defeated and we are made free. This, too, is another aspect of atonement (known by theologians as Christus Victor), where Messiah's ultimate goal and purpose was the defeat of Satan and sin and death through the power of His resurrection.
Let us take this one more step. Colossians 3 says that we have "put off the old self with its practices and have put on the new self, which is being renewed in knowledge after the image of its creator." You see, back in the garden, we were given one task--to reflect the character and glory of God as His image-bearers in creation. Adam failed, and through him we all failed. Then, four thousand years later, Yeshua took up this mantle and did what the first Adam could not do. He became the second Adam, the perfect image-bearer (this fulfillment of Adam's role we call recapitulation). As we shed our our old selves and are made new in Him, we are molded into His image that we might be restored to that original role. That we might once again reflect the character and the glory of our Creator.