Contained in today's reading is one of the most explicit explanations of what happened on the cross.
In 5.21, God made Jesus who knew no sin to be sin. Why did He do that? How did He do that? Good questions. The how is by placing our sin onto Jesus. 1 Peter 2 says that Christ bore on sins in His body. When Christ was hanging there, He was bearing my sin like an animal bore a load. He carried my sin, though He didn't deserve it. He carried my sin, though He didn't have to do so. He carried my sin, though He shouldn't have.
And the why? Because by doing so, God could impute the right-standing nature of Christ to me (Bible word: righteousness). Now, because Christ bore my sin, I stand with Christ and because of Christ as rightly related to God.
Martin Luther called this verse the description of the Great Exchange: Christ takes my sin and gives me His righteousness.