Scout does not want Boo to be talked about in the neighborhood if anyone sees her holding his hand and leading him like a child.
Scout has been used to thinking of Boo as a monster, and then a phantom. When she finally meets him, she sees how timid and fragile he is. She knows that communicating with her is difficult, and she refers to him as childlike and pale.
Scout is unsure about how to deal with Boo Radley. She has never felt comfortable going near his house, and taking him home is a surreal experience for her. Yet it is important to Scout to maintain his dignity. She does not want anyone to see her leading him around like a child. So she comes up with a solution.
He had to stoop a little to accommodate me, but if Miss Stephanie
Crawford was watching from her upstairs window, she would see Arthur Radley escorting me down the sidewalk, as any gentleman would do. (ch 31)
Scout makes it look like he is leading her, rather than she is leading him. That way if anyone happens to see them, Boo Radley will not seem vulnerable and it will not be obvious she is taking the shy man home.
This demonstrates Scout's maturity and ability to empathize. Difficult as it was for her in the beginning, Scout has learned how to walk inside another's skin.
After Boo Radley saves the lives of Jem and Scout as they are walking home from the Halloween program at school, he comes into the Finchs' home. He sits quietly in the corner in Jem's room and later on the porch. Just by sitting quietly with Boo and observing him, Scout gains more empathy for the man she has feared and, in some ways, insensitively mocked. Without speaking, Boo stands and nods toward the front door, and Scout understands this means he wants to see Jem again. She leads "Mr. Arthur" through the house and into Jem's bedroom. Boo "had drifted to a corner of the room," but after Aunt Alexandra leaves, Scout takes him by the hand and leads him to Jem's bed. She tells Boo it's okay to "pet" Jem. He squeezes her hand, and she knows that means he wants to go home. She then takes him by the hand and leads him back downstairs.
At the door to the front porch, it becomes clear that Boo doesn't want to go to his home alone. He has rarely left his house, so to make that trek, even shrouded in darkness, makes him anxious. He leans over to Scout and whispers, "Will you take me home?"
This puts Scout in the position of being the "mature" person with someone who is much older than she is. She understands his vulnerability, but she also knows how gossip flies in their small town. She herself has spent hours speculating and making up stories about Mr. Arthur. Now, in a dramatic turnaround, rather than having fun at Arthur's expense, she puts herself in his shoes in a way that is astonishing for a child of her age. She wants to protect his reputation--even more than he does. So rather than "lead him" down the street to his house, she instructs Mr. Radley to crook his arm, and she grasps it around the elbow, allowing him to escort her in a posture that would be culturally appropriate for a man to escort a lady. Again, this is remarkable for Scout because she has often resisted ladylike behavior.
Her statement that she would lead him through the house but not lead him home shows that she has come to understand and appreciate that the way one must behave in society may differ from the way one behaves in the privacy of one's own home. She adopts the norms of ladylike behavior not to make herself look good but to protect Mr. Radley from unkind gossip. In doing so, she demonstrates the empathy that her father has been trying to develop in her.