What is the idea behind animal sacrifice?

This in general. Please don’t poke fun at religion. This is something which I find most perplexing. Animal sacrifice has been a part of many religions, so there must be some general reason behind it. Is it like a trade with the gods? We give this animal to you so that you will treat us nicely. Is it an expression of devotion? We slaughter our best livestock to show our love and respect. In some cases the sacrificers eat most of the sacrifice, which would seem to eliminate the previous explanations.

Christians say that God sacrificed his only son. I mean no offense, but that makes no sense to me. Who was Jesus being sacrificed to? Is this a kind of reversal of the usual process. Instead of people making a sacrifice to God, did God make a sacrifice to the people?