Why can't God just forgive sin?
My answer to this would be that there are two kinds of forgiveness, one of which is a lot more 'powerful' than the other, and God needed to use this second, more powerful kind of forgiveness. Moreover, giving this kind of forgiveness required Jesus to die on the cross.
Imagine a thief who keeps stealing some guy's stuff - let's say John's stuff. John is so nice that whenever the thief steals from him, he forgives the thief. But the thief never changes his behaviour. John can forgive the thief all he wants, but it doesn't stop the thieve from stealing. Forgiving the thief doesn't make the thief a better person.
John's kind of forgiveness could be called the first kind.
The story shows that John's kind of forgiveness doesn't do that much. John's forgiveness won't make the thief stop stealing, it will only prevent John from seeking justice and might also relieve some emotional tension from his anger. John's kind of forgiveness won't change the thief's behaviour.
If God's forgiveness is like John's forgiveness then God's forgiveness won't change people's behaviour. If God's forgiveness is like John's forgiveness then we'll act in heaven the way we do on earth. This could lead to heaven having such things as people really disliking one another, splits between different groups, cliques, and so on. Not really a great picture of heaven.
The Christian idea is that to solve humanity's problems, God needed a more powerful 'second' kind of forgiveness - one that changes behaviour. That's the kind of forgiveness you need to really deal with humanity's issues.
See Col 2:13: "You were dead because of your sins and because your sinful nature was not yet cut away. Then God made you alive with Christ, for he forgave all our sins."
The Bible says that when God forgave us He managed to change our behaviour as part of the forgiveness. Our sinful nature was 'cut away' by God's forgiveness, although we will still fight against it until Jesus comes (Gal 5:17).
Imagine John forgiving the thief with such 'power' (somehow) that the thief decided never to steal again! That would be similar to the second kind of forgiveness.
So how does it work?
The Bible says that the mechanism for God's more powerful kind of forgiveness must involve Jesus dying for us (Matt 26:39). I'm not too clear on the details of how it works, but I suspect it involves some kind of exchange between sinners and Jesus. 1 Peter 2:24 says, "He personally carried our sins in his body on the cross so that we can be dead to sin and live for what is right. By his wounds you are healed", in Romans 6:6, "our old sinful selves were crucified with Christ", and in Gal 2:20, "It is no longer [my old sinful self] that lives, but Christ lives in me".