I don’t get it. You say it is important to “give the other a gift” when we forgive. Why the emphasis on gift-giving?
I emphasize gift-giving because that is part of the definition of what forgiveness is. As we forgive, we give to the other, particularly the one who has hurt us. Thus, to give a gift of some kind (perhaps a smile or a returned phone call) is to exercise the virtuous nature of forgiveness.
For additional information, see Forgiveness Defined.