Why must sins be paid for? Why can't God just forgive and forget them? He can't because it would be wrong to do so. Sins must be paid for because sin is breaking the law of God and if God ignores his own Law then he would be approving of sin. To ignore his own law would be to contradict his holy character from which the Law is derived. But, God cannot ignore his own holy character (1 Peter 1:16). Therefore, the Law must be upheld and carried out along with the punishments it contains. In Christianity, the punishment is taken care of one of two ways, by Jesus on the cross or by the unbeliever in hell forever. Either way, the law is satisfied.
Since sin is breaking the law of God, sin is a legal problem, though not only a legal problem. 1 John 3:4 says that sin is lawlessness. The famous prayer uttered by Jesus where he says "Our Father who is in heaven" is found in two places in the Gospels, Matthew, and Luke. In Matthew 6:12 Jesus says forgive us our debts. In Luke 11:4 he says forgive us our sins. So, Jesus equates sin and debt. Since God is holy and cannot ignore sin, then it must be dealt with. The law must be satisfied.
But legal debts can be transferred. I could write a check to you for $100 and when you deposit it into the bank, there is a legal transference of money. Likewise, debts can also be transferred. Sometimes banks buy debts owed by people to various organizations and the debt is then transferred and owed to the bank.
Likewise, in Christianity, our legal sin-debt to God is transferred to the person of Jesus (1 Peter 2:24). He bore our sins in his body on the cross. This way, Jesus pays for our sins and satisfies the requirement of the law by dying on the cross. The Bible says that the wages of sin is death (Romans 6:23). But, for all who do not believe in Christ their sins are not paid for by him. They must satisfy the requirements of the law on their own, in hell.