Many other bloggers have written about forgiveness. There are whole web sites, books and even religious doctrines devoted to the concept, so I won't go into great detail here.
In Christianity, we are asked to forgive those who trespass against us, as we are forgiven our own trespasses (as the Lord's prayer more or less puts it). That is basically asking us to live and let live.
Forgiveness as a way to move on from a bad situation seems to me to be something that has occurred fairly recently. The parents of a murder victim forgiving a murderer, for example, I believe is a fairly new development in our western societies, new as in just a couple of decades old.
Those who have written about forgiveness, who have discussed it and taught it and promoted it and done it, say it is not about forgetting or ignoring the harm another has done you, but about not letting it create (any more) bad feelings in you. Forgiveness is really about no longer picking the scab off a wound, but just leaving it alone.
Let me use that metaphor a bit more: Past hurts, whether they be from parents who abused us, or a spouse who had an affair, or a best friend who turned out not to be a friend, etc., stay painful because we keep reminding ourselves of them. One reason we keep getting reminded is if the person who hurt us hasn't owned up to their part in the situation. But another reason we keep getting reminded, is because we indulge in the pain.
There can be many reasons for why we hang onto a pain for so long. After a while, especially with childhood hurts, it seems that it's just who we are. It's always been this way. We've always felt this way. It's always been this hard, this raw, this upsetting. It's hard to let go because we've let it define us.
Forgiving is about looking at the wound you have, and realizing that the person who initially gave you that wound is not the same person picking the scab off it, making it bleed again. That person is yourself. You are picking at yourself. You are keeping the wound from healing.
An example of forgiving someone can be what I wrote to my own mother recently: "I love you. I forgive you. I no longer blame you for any negativities in my life. Those are my responsibilities." Basically, I forgave her because I finally grew up. I became an adult. I no longer hold another adult accountable for my own feelings and for how deeply and for how long I feel hurt should they hurt me.
Did my relationship with my mother change? Not with that act, no. But now I can think about my mother without anger, without pain, without wanting to hurt her back. And it is very liberating.
So what changed are my feelings. Nothing else. And that is the point of forgiveness: Not to let the other person off the hook, but yourself. Get yourself off that hook of pain and stop struggling.