How the Atonement Works
Here is an explanation of the atonement that works, even if you don’t believe in God or Christ, and if you do, it still works as an explanation.
Why do we think we need a savior? We need to have the ideal of somebody so big and powerful enough and smart enough who has earned the right, through His sacrifice, to forgive us of our sins. The biggest problem we have is forgiving ourselves, but if a being like I just described can forgive us, then we can forgive ourselves.
Christ suffered and died terribly that He might draw all men unto Himself, or in other words, to get our attention.
It is literally our faith in such a being that saves us.
Here’s the extra added part for people who do believe in God. Each of us is part of God. Even atheists have a conscience and a sense of right and wrong. That’s a part of God living in you, whether you choose to call it God, the universe, or whatever. Sin separates us from the rest of God, or from your conscience. If you can’t forgive yourself, then God can’t forgive you. As you judge others, so you will be judged.