From my understanding of various Catholic explanations of The Incarnation.
Firstly, let's acknowledge the mystery. We can talk somewhat intelligently about it. But, the real mechanics, the true significance, the fullness of the sacrifice -- mystery.
That said ...
It's important to remember that Christ is the hypostatic union of God and man. Any interpretation of Christ that doesn't acknowledge that He is both fully human and fully God is not a Christian interpretation. And this hypostatic union is important in every detail of Christ's life. Because Christ is God, anything and everything that Christ does is Godly. And, everything that is done to Christ is done to God.
Hence, we can speak about the significance of Christ's death in two meaningful ways (and probably more).
Firstly, Christ brings God into human judgement, rejection, hate, mockery, suffering, and death. Hence, God is present as God in all of these human experiences, including death. Thus, a human death is "no longer" simply to die, it is to be like God. And since God never dies, neither then do we fully die. And if that isn't enough, in the resurrection, God brings life back into death in a physical manner by rising bodily from the dead.
Secondly, Christ turns God into the very real and incarnate target of our judgement, rejection, hate, mockery, suffering, and murder. Therefore, in judging, rejecting, hating, mocking, torturing, and murdering Christ, we are doing so to God Himself. Christ makes our hate for God a physical manifestation. The passion and death of Christ is the visible and sanctifying incarnation of God's choice to let us hate Him to the greatest extent possible -- unto death. And the resurrection, of course, is God's visible, incarnate choice to return to us, despite our very real hate.
So, what is God sacrificing? He's literally sacrificing Himself. The human aspect of this, which can be reasonably equated with the suffering and death of any other human, isn't really the significant part of this event in and of itself. The human aspect is "merely" the incarnation of a more significant, spiritual reality. God, the creator of all things and our Father, willingly puts Himself at our mercy. He sacrifices His rightful role as the God we love and becomes the God we hate in the very same act by which He brings His own divinity to our suffering and death.
(I suspect most parents (and grown-up children) readily recognize this sacrifice in a limited, imperfect fashion.)