How do the Stoics say to forgive people?

The Stoics do not treat forgiveness as a favor you grant once you have decided someone deserves it. They treat it as the correction of a mistake in your own thinking. Once you see the mistake, the anger that needed forgiving in the first place tends to disappear.

Start with Epictetus's account of why people wrong you. In the Handbook he writes that when someone harms you or speaks badly of you, remember that he acts or speaks from a supposition of its being his duty, and that it is not possible for him to follow what appears right to you, but only what appears so to himself (Handbook 42). If he is judging from a false appearance, Epictetus says, he is the person hurt, since he too is the person deceived, the way a false proposition is not harmed by being mistaken but the person who believes it is. From this he draws a plain conclusion: you will meekly bear a person who reviles you, because in every case you can say, "It seemed so to him" (Handbook 42).

This is the Stoic root of forgiveness: wrongdoing is a failure of judgment, not a straightforward exercise of a bad will against you. In the Discourses, Epictetus pushes this further with the case that offends people most, calling someone a thief. He asks what you mean by that. They are mistaken about good and evil, he answers, and asks whether we ought to be angry with them or to pity them (Discourses 1.18). He rejects the idea that such people should simply be destroyed, comparing that impulse to asking whether we should destroy a blind or deaf man, since the wrongdoer is blind not in the faculty that distinguishes colors but in the faculty that distinguishes good and bad (Discourses 1.18). His conclusion is direct: pity him rather, and drop this readiness to be offended and to hate (Discourses 1.18).

Marcus Aurelius gives the fullest working method. In Meditations 11.18 he lists considerations to run through when someone has offended you. We are made for one another, so anger against a fellow being cuts against our own nature. People act as they are compelled to by their opinions, often with a kind of pride in what they are doing. If they act rightly, there is nothing to resent, and if they act wrongly, they do so involuntarily and in ignorance, since no soul is willingly deprived of the truth (Meditations 11.18). He adds a harder check on your own standing to judge: you too do wrong, and where you abstain, it is often from cowardice or concern for reputation rather than pure excellence (Meditations 11.18). And he ends with the discipline underneath it all: it is not other people's acts that disturb you, but your own opinions about them. Take away the opinion that the act was something grievous, and the anger goes with it (Meditations 11.18).

The common misreading is that this is forgiveness as a moral achievement, a generous thing you do for the wrongdoer. Read closely, it is closer to accuracy. You are not excusing the act or pretending it was harmless. You are correcting your judgment about what actually happened, so that you see a person deceived about good and evil rather than an enemy who deserves your hatred. Marcus is explicit that this does not mean acts are all equally acceptable: elsewhere he calls acting unjustly a kind of impiety against the rational nature that made people for each other (Meditations 9.1). Seeing wrongdoing as ignorance does not mean it stops being wrongdoing. It means your anger, which rests on the false belief that the wrongdoer's inner ignorance is itself an injury against you personally, has nothing left to stand on.

What changes in practice is the sequence you run when someone wrongs you. Instead of asking how dare they, ask what error they are under, whether they can be shown it, and whether your own judgment of the act, not the act itself, is what is disturbing you. That is the concrete move: before you answer anger with anger, name the opinion doing the damage, and check whether it is one you actually need to hold.