Interesting enough, there is no definitive theory among Christian theologians as to why Jesus died or how his death saves us. That's surprising when you consider the amount of ink used on other topics. My personal view is that this begins to get into areas that we can't know. We know what God has done for us, but we don't know what alternative were available and why he chose to do things the way he did. In the OT there seem to have been few of the kinds of questions that bother later Christians about Jesus' sacrifice.
Nobody asks why sacrificing an animal accomplishes anything.
I suspect originally, the question of who Jesus' sacrifice is to would not have occured to anyone. In both what Jesus is quoted as saying and other NT writers' reflections, the sacrifice is clearly seen as being on behalf of those who believe in him. Hebrews is particularly full of this sacrificial language. However there's no real explanation of how a sacrifice helps us. In Paul's letters it is explained that Christians are united with Christ in such a way that they experience both Christ's death and resurrection.