What Do We Mean When We Say Jesus Died for Our Sins?

What does it mean to say that Jesus died for our sins? How precisely does his cross save us? The first Christians saw sin as a sort of imprisonment, like being held for ransom, and in the dying and rising of Jesus, they experienced freedom. What freed them was God's solidarity with them in their fear, even their fear of death. How do you experience the power of Jesus' death on the cross? How does it set you free?