Christians: Why is sacrifice necessary to atone for sin?

In the old testament, the Jewish people had to sacrifice pigeons or barnyard animals to God before their sins could be forgiven. How does an innocent unblemished animal dying make up for me stealing something? Wouldn't it be better to return the stolen item, or pay for it and apologize? I've heard the argument that God's perfect justice requires all sins to be punished and that the penalty of sin is death, so death is necessary for the sin to be atoned for. How does the death something else make up for MY particular sin though? I asked this question of Chrisitans because this belief is a central part of Christianity...Jesus, playing the role of the innocent perfect sacrifice, dies to take the punishment for all of our sins once and for all. But how does Jesus' dying allow me to be forgiven? Personally I think this idea of substitutional atonement is akin to the Aztec idea of killing people to appease the sun god. This belief was one of the reasons I decided to not be Christian.