So, I do, and it's informed by religion, but I'll try to phrase it as LW-friendly as possible: to free somebody else of claims I have against them.
It's not an emotional state I enter or something self-centered (the "I refuse to ruminate about what you did to me" pop song thing), though sometimes it produces the same effects. The psychological benefits are secondary, even though they're very strong for me. I usually feel much more free and much more peaceful when I've forgiven someone, but forgiveness causes my state of mind, not vice versa. It's like exercise: you did it and it was good even if you didn't get your runner's high.
Other useful aspects, from the most blandly general perspective: it's allowed me to salvage relationships, and it's increased the well-being of people I've forgiven. I've been the beneficiary of forgiveness from others, and it's increased my subjective well-being enormously.
From a very specific, personal perspective: every time I experience or give forgiveness, it reminds me of divine forgiveness, and that reminder makes me happier.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.