The opening quote to this chapter sums up why you would reach for forgiveness. Repeated here:

If we could read the secret history of our enemies, we should find in each man’s life sorrow and suffering enough to disarm our hostility.

Forgiveness is something we do for ourselves. By forgiving others, we free ourselves from resentment and anger, from the weight of the hurt they cause you. When there is something to forgive, when there is some sort of damage done to you, not forgiving the other person, keeps the damage alive and burning within you.  

Of course, in close relationships it’s not just a matter of that weight if the relationship is active, by not forgiving the other person you prevent the relationship from being the best it can be. That affects how you behave toward the other person and that person toward you.