How does Scout forgive Jem in To Kill a Mockingbird?
You don't specify which time Scout found herself forgiving Jem, but usually it is because she loves her brother and the two are practically inseparable.
Jem antagonized me sometimes until I could kill him, but when it came down to it he was all I had. (Chapter 11)
She forgives Jem for deserting her at school because he explains that "school's different" from home. She forgives him when he pushes her in the tire onto the Radley steps in part because she realized he "was offended by my contradicting him on Hot Steams." She forgives him when he returns to the Radley fence to retrieve his lost pants because "it was no use" trying to stop him. When "Jem stayed moody and silent for a week," Scout forgave him by trying to "climb into Jem's skin and walk around in it." When Jem destroyed Mrs. Dubose's camellias, broke Scout's baton and then "yanked my hair," she forgave him because he "looked like he was sorry. There was nothing to say." When Jem's moodiness caused Scout to think he had a tapeworm, she forgave him when Calpurnia explained that Jem was growing. Finally, she forgave Jem for ratting out Dill to Atticus--when he
... broke the remaining code of our childhood...
--after they found Dill under Scout's bed after he had run away from home. She saw that Jem was right about Atticus knowing what to do, and
Since things appeared to have worked out pretty well, Dill and I decided to be civil to Jem. Besides, Dill had to sleep with him so we might as well speak to him. (Chapter 14)