How does Jesus' sacrifice on the cross lead to the salvation of humanity or anyone?
Except for the book of Hebrews, which strongly connects the Christian story to the temple system, not much elsewhere in the NT delves into what the mechanics of salvation are. It is, however, made clear that it was necessary. Paul speaks routinely of "Christ crucified" as well as "if Christ be not raised, we are of all men most miserable."
There are three theories of salvation that I can remember off the top of my head. The first is the theory of substitutionary atonement. That is, as we are sinners we have sin that requires atonement. God, as a type of judge, has condemned us. Jesus then volunteers to take our punishment and this pleases God who then, after the sacrifice, imputes Christ's righteousness to us. This imputation is almost always considered to be conditional. We don't get the imputation without belief.
Some Christian scholars complain that this doesn't make sense, for reasons I'll let go. So secondly there is what is known as Christus Victor. Essentially, Jesus conquers. He conquers both (and 'both' is essential) life as well as death. You could say we are imprisoned both by life and eventually by death. Christ knocks down the prison walls. Christ's death on the cross is obviously necessary for the second part--the conquering of death. The conquering of life, so to speak, is covered by Jesus' example of how to live on earth.
A third theory is propounded by C. S. Lewis. (It may not have been original with him, but it is from Mere Christianity that I learnt it.) This is called the Perfect Penitent. Because we are sinners, we need to repent. However due to the corruption of sin, we are incapable of the appropriate repentance. Jesus became human to be the perfect penitent. To repent for men in general since we could not. This repentance carried out both by his life and death.
Brian McLaren's Generous Orthodoxy (which I highly recommend) lists at least two more theories, IIRC. But the above, I think is representative. I apologize however if I over simplified.
In my experience, most Christians subscribe to the idea that our sin requires payment and that Christ paid it. His perfection and perfect death is sufficient to cover all those who want it. For universalists, it covers all. For others, it is sufficient to cover all but is only effective to those who desire it. To a strict Calvinist, Christ's death pays only for the elect.
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