Religion doesn't have a lock on forgiveness. Not to say libertarian folks aren't or can't be religious or spiritual. Forgiveness doesn't just exist in the realm of the divine.
People are inherently flawed. Again, not just from a spiritual point of view but biologically and mentally as well. We have literal biological faults within us that cause us to do things irrationally or entirely out of our control.
Mentally, our thinking is affected by education, emotions, again biology, and beliefs.
Forgiveness for libertarians is the ability and willingness to recognize that no one is perfect, all people fail and that just because people fail doesn't necessarily make them a bad person.
It means being able to say, "I've done plenty of things I'm not proud of in my life and I expect to do things in the future that are wrong, maybe bad, and yet I intend to continue to make the effort to be a good person."
If you can admit that you yourself will fal, aren't perfect and hope that people can see the good person you mean to be regardless, then you will be more likely to do the same for other people.
To forgive is to identify, accept, evaluate, and address. To forgive myself and forgive others. It's not something that requires a deity or a priest in day to day life.
I would love to be my ideal self. Patient, tolerant, wise, forgiving, concerned, empathetic. I'm almost never all those things at once. I'm rarely some of those things ever. I try to be as many of them as much as I can. At least, most of them.
I really lack concern and empathy for others. I know, I know...
But I forgive myself and I forgive others. I have bad days, I "slip", I can be mean and ugly especially when I don't feel good.
Sometimes people say things they don't mean in the heat of things, I can forgive that.
Sometimes people don't have all or the right information and argue anyway, I can forgive that.
Sometimes people take shortcuts under stress,I can forgive that.
There re lots of things I can forgive in myself and others. When it comes to things I can't forgive. I am hardest most especially on myself. I try to hold myself to certain standards, when I violate them, I find it hard to forgive myself. Sometimes I don't. Sometimes I just have to suck I up and use that experience as motivation to not let it happen again because of the grudge I bear, even against myself.
Ultimately, forgiveness is relating to others and knowing we share the state of being known as imperfection.