The last three verses of Isaiah 53 give the spiritual significance of what’s happened at the cross, the purpose of God which was accomplished.
. . . when you shall make his soul an offering for sin, he shall see his seed, he shall prolong his days, and the pleasure of the Lord shall prosper in his hand.
It’s also possible to translate it “when his soul shall make an offering for sin.” Jesus’ soul was made an offering for sin. Jesus’ body bore our pains, griefs and sicknesses, but his soul was made the sin offering.
Now in order to understand the implication of that, you have to be just aware of the procedure for the sin offering in the Old Covenant under the Levitical law. When a man sinned he had to bring the appropriate offering. It might be a sheep or a goat or a bullock. When he arrived the priest laid his hands on the head of the offering and the man confessed his sin over the offering. And symbolically the sin of the man was transferred to the goat or the bullock or whatever. Then the goat or the bullock paid the penalty for the man’s sin. Instead of killing the man they killed the goat. So the sin offering was something to which the sin of the person was transferred so that the offering paid the penalty for the man’s sin. See the picture?
Jesus identified with the sin of the whole human race. Our sin was transferred to him and then he paid the penalty in our place. So his soul was made the sin offering. He was identified with our sins.
Now if the priest had simply confessed his sin, and laid his hands on the head of the bullock, that would not have resolved his problem. The next thing he had to do was kill the bullock. And when he killed the bullock, he killed his own sin. He identified himself with God’s judgment on his sin. Repentance is killing the bullock.
Paul is saying the same thing in 2 Corinthians 5:21 that Isaiah had said in Isaiah 53:10. In fact, he is saying: God the Father made him, Jesus the Son, to be sin for us, that we might become the righteousness of God in him.
He was made sin with our sinfulness that we might become righteousness with his righteousness!