I've never completely understood the purpose as to why God required that the Israelite's should kill an animal in order for God to forgive them.
There were two main meanings to this. The first was as a symbol of Christ, whose blood makes it possible for everyone to have their sins forgiven. The second is as a sacrifice in the more traditional sense: willingly giving up something important and valuable. And when your animals are both your livelihood and the food on your family's table, giving the best of your flocks to the Lord (and the sacrifices were required to be the best, not just any old animal) is quite a sacrifice indeed! By the Law of Sacrifice, the Lord teaches us to have faith and trust in him instead of getting too attached to material possessions and trusting in them to bring us happiness and success in life.