It is towards the end of Chapter 28 of Harper Lee's To Kill a Mockingbird that Scout witnesses Arthur (Boo) Radley rescue her and Jem though she doesn't understand what she is witnessing until much later.

Towards the end of Chapter 28, approximately 4 pages from the end, Scout realizes a fourth person, aside from she, Jem, and whoever is attacking them, has appeared under the tree. She hears one man cough "violently, a sobbing, bone-shaking cough" and another man breath heavily. She then hears the man who was breathing heavily grope along on the ground, "searching for something," and begin to "pull something heavy along the ground." She feels along on the ground for Jem but only finds a bearded man who smells of whiskey lying there. When she looks toward the street light, she sees a man carrying Jem, staggering under the heavy load. The strange man and Jem are the first to reach the Finch's home, and Jem is immediately carried into his room. Chaos erupts inside the Finch household as both Dr. Reynolds and Sheriff Tate are notified. It's not until after Dr. Reynolds arrives and examines both Jem and Scout that Scout is able to go into her brother's room to see him. Once in Jem's room, she sees for the first time the man who carried Jem home but only recognizes him as "some countryman [she] did not know."

It is not until the final page of Chapter 29 that Scout realizes who the man is and who rescued her and Jem. In Chapter 29, she recounts the events of the attack as she remembers them to Atticus and Sheriff Tate in Jem's room while he sleeps. As she recounts events and turns to the man in the room for his own version, she notices the whiteness of his skin, sees his nervousness, and sees him timidly smile at her. It's at this moment she realizes she is seeing her neighbor Arthur for the first time, and the realization brings her to tears. As he smiles at her, she describes that her "neighbor's image [became] blurred with [her] sudden tears." These tears are tears of gratitude and a certain sign she has finally shed all of the false assumptions she had developed about Arthur and has come to feel affection for him; she now sees him as the children's savior, which allows her to see him as the tender, caring, albeit reclusive, man he truly is.

Though it took Scout nearly the entire book to develop affection for her neighbor Arthur, Jem develops affection much sooner since he is quicker to understand things than Scout due to his age. Jem develops affection for Arthur the moment Jem realizes that it is Arthur who has been leaving the children gifts in the knothole of the oak tree on the Radleys' property. Scout and Jem decide to leave a thank-you note to whoever is leaving the gifts, and Jem is devastated to find that Arthur's brother Nathan had filled in the hole with cement. Jem is devastated because not only does he no longer have any means of expressing his gratitude, he also has no means of making amends for having mocked Arthur. In fact, Jem is so devastated that he is moved to tears, as Scout notes in the following:

He stood there [on the porch, looking towards the Radleys' property] until nightfall, and I waited for him. When we went in the house I saw he had been crying. (Ch. 7)

Jem's tears, just like Scout's tears later, are a certain sign that Jem early on had begun to see Arthur as a kind and caring person and develop affection for him.

In the first few pages of Chapter 29, Scout begins treating Boo much better, calling him "Mr. Arthur" rather than "Boo" and inviting him to sit on the rocking chair on the front porch. Knowing Arthur is not used to bright lights, Scout leads him to a spot in the shade: 

Feeling slightly unreal, I led him to the chair farthest from Atticus and Mr. Tate. It was in deep shadow. Boo would feel more comfortable in the dark. 

By the end of Chapter 30, Tate convinces Atticus that Arthur killed Bob Ewell and that it would be unfair and unnecessary to put Arthur through a trial or any more public exposure in order to conclude the nature of Bob Ewell's death. They agree to conclude that Bob Ewell fell on his own knife. Atticus asks Scout to understand why they've come to this conclusion and Scout replies that putting Arthur through something like this would be akin to killing a mockingbird: in other words, it would be unjust to hurt someone who has done nothing wrong. 

In Chapter 31, Scout begins to see Arthur more like a guardian angel than the monster the children and gossips had made him out to be. She also understands the significance of considering things from the perspective of others. She now cares enough to really consider Arthur's view of the world: 

Summer, and he watched his children’s heart break. Autumn again, and Boo’s children needed him. 

Atticus was right. One time he said you never really know a man until you stand in his shoes and walk around in them. Just standing on the Radley porch was enough. 

Boo Radley comes to the rescue of Jem and Scout in Chapter 28 of To Kill a Mockingbird (pages 262-263 in my old paperback copy from 1982). Scout is never actually able to see Boo, since her ham costume obscures her view, but she does realize that "It was now slowly coming to me that there were four people under the tree"--Jem, Scout and the two unidentified men (Boo and Bob Ewell). When Scout later sees Boo hiding in the shadows of a corner in Jem's bedroom, her fantasy of finally seeing Boo comes alive in a most unexpected way: Boo has saved her life from the murderous hands of Bob Ewell (Chapters 29-30). After walking Boo home in the final chapter, Scout reflects about their meeting while standing on the Radley porch.

Boo was our neighbor. He gave us two soap dolls, a broken watch and chain, a pair of good-luck pennies, and our lives. But neighbors give in return. We never put back into the tree what we had taken out of it: we had given him nothing, and it made me sad.  (Chapter 31)