If someone offends you he usually gets blamed for the consequences. You can hold it against him which tends to drag you down. But if that person says he is sorry you should forgive him. For serious crimes many people never forgive the offender. However, Christians tend to follow the Bible's teaching and forgive the perpetrator. "'Vengeance is Mine, and retribution'" (Dt 32:35). How does God forgive our sins? Zecharias prophesied concerning this (Lk 1:67) saying that you become aware of "the knowledge of salvation by the forgiveness of sins" (:77). Jesus "will save [i.e. salvation] His people from their sins" (Mt 1:21). It starts with "'repentance for forgiveness of sins" (Lk 24:47). Then, "'I will forgive their iniquity and their sin I will remember more'" (Jer 31:34). Sin had to be destroyed and the sins produced by it had to be accounted for. "Your sins will be wiped away" (Ac 3:19). John saw the "Lamb of God who takes away the sin of the world" (Jn 1:29). Does that mean that the world's slate is wiped clean and that's the end of it? The only way forgiveness can be received is if God's work is appropriated. Sins are committed in the flesh, so at the basic level, God had to "put to death the enmity" (Eph 2:16) and "condemned sin in the flesh" (Ro 8:3). It was done so "the body of sin might be done away with, that we should no longer be slaves to sin" (6:6).
How is this accomplished? It is because "Christ died for our sins" (1Co 15:3). He "was delivered over because of our transgressions" (Ro 4:25). Jesus said, "'I DESIRE COMPASSION, AND NOT SACRIFICE', for I did not come to call the righteous, but sinners" (Mt 9:13). "God demonstrates His own love toward us, in that while we were yet sinners, Christ died for us" (Ro 5:8). He gave "His life a ransom for many" (Mt 20:28). He was "offered once to bear the sins of many" (Heb 9:28). He "gave Himself as a ransom for all" (1Ti 2:6). It results in "the forgiveness of our trespasses, according to the riches of His grace" (Eph 1:7). He is the "propitiation for our sins" (1Jn 2:2).