Scout’s decision to call Boo Radley “Mr. Arthur” shows that she has matured, because she sees him as a person and not a myth.
During the first few chapters of the book, Scout and the other children see Boo Radley as a neighborhood myth. He is spooky and silly, and more of a legend than a real person. The children play games where they act out his “story” and try to make him come out. They are very curious about the infamous and mysterious Boo Radley, who was a troubled child and apparently did have a run in with the law.
Jem’s “reasonable description” of Boo Radley is a perfect example of how the legend of Boo is larger than life and kind of silly.
Boo was about six-and-a-half feet tall, judging from his tracks; he dined on raw squirrels and any cats he could catch, that's why his hands were bloodstained- if you ate an animal raw, you could never wash the blood off. (Ch. 1)
As Scout gets older, she starts to outgrow the Boo Radley legend and begin to see Boo as a person. She learns his story, and his name (Arthur). She begins to feel sorry from him. Jem, who is older, realizes what is happening sooner. When Boo begins to watch the children and leaves presents from them in the tree, Jem puts two and two together. He sews Jem’s pants together, and puts a blanket on Scout’s shoulders at the fire. Boo Radley is looking out for the children.
The whole thing comes to a head after the trial, when Atticus defends Tom Robinson and makes it clear that Mayella was not even raped. This makes Bob Ewell look like a fool, and makes him awfully angry. He decides to get revenge on Atticus by attacking his children, but Boo Radley is there to protect them.
Boo defends Scout and Jem, and kills Bob Ewell. He follows the children home, and while everyone is arguing about what to do they realize that Arthur Radley is still there. Eventually, it is agreed that they need to protect Arthur Radley at all costs. Scout is old enough to understand what has happened enough to know that Arthur is the reason she is alive. She treats him with respect, dignity, and kindness.
People have a habit of doing everyday things even under the oddest conditions. I was no exception: "Come along, Mr. Arthur," I heard myself saying, "… I'll just take you to the porch, sir." (Ch. 30)
Scout realizes that Arthur Radley is her mockingbird. He is a delicate creature victimized by society for years and years for reasons beyond his control. His crazy family made him the way he is, and either forces him to be a recluse or makes him want to be one. She understands that he needs to be handled with care, and she does so. This demonstrates a great deal of maturity and empathy on her part. She has taken to heart Atticus’s lessons about mockingbirds and walking around in other people’s skin.