The death of Christ is often explained either as a debt being paid - that is, His death paid the price of our sin - or as one innocent person dying in the place of other guilty people who have been condemned to die (that is, as a substitute). These are two different metaphors, but they often get confused and used together in explaining the 'atonement', or how Christ's death brings about our salvation. It's one thing to speak about a 'debt' being forgiven, but to then mix this up with a capital punishment for a criminal offense would be to confuse the metaphors.
If we stick to the language of debts being paid then Jesus must have paid the debt to someone - if indeed He paid a debt. This is quite different from someone dying as a substitute in place of another for a crime.
Paul used a variety of metaphors from the marketplace, the slave trade, the law courts and the Temple, because no one analogy is adequate or complete in itself. No one metaphor was adequate for him, and no metaphor should be pushed too far.
However, Jesus rarely, perhaps never, spoke of His death as an 'atonement'. The Gospels record only one brief saying which possibly alludes to His death as an atonement - the 'ransom saying' of Mark 10:45 (parallel Matt 20:28), which may, or may not, be a reference to His death. Jesus' references during the last supper to His blood being shed to seal the new covenant is the language of covenants, not atonement. So it's actually doubtful whether Jesus ever referred to His own death as an atonement.
On the other hand, Jesus spoke frequently of God's forgiveness, His abundant generosity, and His graciousness. There is nothing in any of His parables, stories or sayings which suggests that a price of any kind had to be paid to secure God's forgiveness. The stories which refer to debts being forgiven all emphasise the undeserved kindness shown by the one forgiving the debt. If any debt was owed by Adam or his descendants because of his sin or theirs, then the debt was owed to God. If Jesus death was to pay a debt then the debt must have been paid to God, and that would put God in the position of demanding the death of His own Son in order to satisfy a debt to Himself. The other alternative would be Anselm's satisfaction theory which had the debt being paid to the devil, which I personally think is absurd.
If Jesus died as a substitute, taking our place for the crimes we have committed, then He suffered the punishment for our sins which was due. There is no need for forgiveness then, because the sentence has been carried out. We are free, not because we have been forgiven, but because someone else took our place.
As I see it, the only way we can understand forgiveness is to see it as a gracious act of God in NOT demanding payment or punishment for our sins. If we use the metaphor of a debt, then the debt is paid and is not forgiven. If we use the language of capital punishment then the sentence has been carried out and the guilty party has a substitute who dies in their place, but the crime is not forgiven. Neither of these analogies explain what actually happened: God chose to forgive our sins even though there was absolutely nothing we could do to merit or deserve His forgiveness, and even though it would be impossible for us to find a substitute who could suffer the punishment which our sins deserved.
As I see it, Jesus' death was a demonstration of how far God's love would go in order to save us, not what God demands in order to be satisfied. Several Scriptures point us in this direction:
Romans 5:8The death of Christ is primarily a demonstration of the love of God. It was not an act to appease an offended deity. It was not a mechanistic or legalistic sacrifice to satisfy the requirements of any religious law. It was not a demonstration of what "the flesh" deserved. It was an act of love.
But God demonstrates his own love for us in this: While we were still sinners, Christ died for us.
The life I live in the body, I live by faith in the Son of God, who loved me and gave himself for me.
Christ loved us and gave himself up for us as a fragrant offering and sacrifice to God.
1 John 3:16
This is how we know what love is: Jesus Christ laid down his life for us.
1 John 4:9
This is how God showed his love among us: He sent his one and only Son into the world that we might live through him.
1 John 4:10
This is love: not that we loved God, but that he loved us and sent his Son as an atoning sacrifice for our sins.
William Barclay puts it very beautifully in The Plain Man Looks at the Apostles' Creed:
"But why the death of Christ? If Jesus had stopped before the cross, it would have meant that there was some point beyond which the love of God would not go, some limit to his love. But in Jesus God says: 'You may disobey me; you may grieve me; you may be disloyal to me; you may misunderstand me; you may batter me and bruise me and scourge me; you may treat me with savage injustice; you may kill me on a cross; I will never stop loving you.' This means that the life and death of Jesus are the demonstration and the proof of the limitless, the undefeatable, unchangeable, unalterable, infinite love of God." (My emphasis).This is the most beautiful summary I have ever read of the motivation beyond the cross.