In Lee's To Kill a Mockingbird, how does Scout help Boo Radley?

In the novel, Boo Radley saves Scout and Jem from Mr. Ewell, and he earns the gratitude of both Atticus and the children.

When Sheriff Tate and Atticus discuss the aftermath of Mr. Ewell's attack and Jem's possible culpability for Mr. Ewell's death, Boo Radley is present. However, because of his shy ways, Boo doesn't join in the conversation. Scout instinctively knows the reason for his reticence, and she's respectful of Boo's quiet ways. With a compassion and sensitivity beyond her years, Scout quietly leads Boo to the farthest corner of the porch, where he will feel more comfortable in the shadows. Scout doesn't judge Boo; she helps him by extending him the gentlest of courtesies, but she doesn't make any effort to change who he is or to cause him discomfort.

After Sheriff Tate and Atticus decide on the best course of action regarding Mr. Ewell's death, Boo gets ready to leave. Before he leaves, however, he nods towards the front door. Instinctively, Scout pipes up, "You’d like to say good night to Jem, wouldn’t you, Mr. Arthur? Come right in." Scout readily anticipates Boo's desires; quietly, she takes Boo by the hand to Jem's bed.

For her part, Scout is able to see Boo for who he is: though a socially-awkward individual, Boo is a man of courage. He put his life on the line to save Jem and Scout from certain death. It was Boo who stabbed Mr. Ewell, an action at once incongruent with his outward demeanor as well as characteristic of his deeper personality. Recall Miss Stephanie's anecdote about Boo, who, when he was thirty-three years old, drove a pair of scissors into his father's leg and then comfortably resumed his scrapbook activities after retrieving the scissors.

Scout realizes that Boo is more complex than most of the adults make him out to be. She definitely understands his tendency to withdraw into himself and makes allowances for it. Yet, she also understands his ability to react in unexpected ways. Her receptivity to Boo can be seen at the moment Boo stands over Jem's bed. Almost imperceptibly, Boo lifts his hand, only to drop it again. Scout instinctively knows what he needs and wants to do, and she gently gives Boo permission to touch Jem.

After touching Jem lightly on the hair, Boo quietly asks Scout to take him home. With great kindness and tact, Scout asks Boo to bend his arm a little so that she can put her hand into the crook of his arm. Scout does this because she wants anyone who's watching to think that Boo is the one escorting her, "as any gentleman would do." Scout essentially protects Boo's dignity and image in the eyes of the community.

So, throughout their interactions, Scout helps Boo by treating him with sensitivity, kindness, and compassion. In Scout's eyes, this is the only way she can repay a man who's done so much for her and Jem.