It is called substitionary death. Sin basically demands death. Both physical and spiritual.
Animal sacrifice is shadow of Jesus' death on the cross. Well, it wouldn't be right to demand the death of a human for our own sins and given the fact no one is perfect enough to qualify for anyone else's substitionary death. Exception being the Son of God.
So when animal was sacrificed under the Old Covenant, basically the idea was one can lay all of their sins upon this animal and while it bears their sin, the animal will substitute its death for their death so in order that their life and repentant heart will be renewed for another year. The animal bore their sins and died their death (both physical and spiritual) in order that theymay live. If there is no substitionary death offered for sin, then with physical death bears the brunt of spiritual death."
Bottom line: sin = imminent phyisical and spiritual death , substitutionary sacrifice = life
Thank you for trying to explain it, but I'm sorry, it makes no sense to me at all.
My own belief system works this way: If I hurt someone else, I should go to them and apologize and work it out with them. The idea that I can kill an animal to atone for that hurt is ludicrous, and an insult to the other person, and counterproductive towards the positive growth of that relationship.
If someone stole some money from me and I found out, and then they said "oh, I slaughtered a goat last week to atone for that" I'd be like, really? What does that have anything to do with repaying me back my money?