Of the two children, Jem is the one who realizes their ignorance towards Boo Radley early on. However, Scout (being only a 6, 7 and finally 8 year old child) doesn't understand who Boo really is until the ending chapters. Her development is seen most clearly when Jem is unconscious in bed with a badly broken arm. Once introduced to Boo (who is in the shadows of the bedroom), Scout takes him and leads him out to the porch --to the darker part of the porch--which shows she is putting herself in his shoes. She knows that he would prefer to be in darkness.
She then expresses her understanding of Boo and the situation he is in when she asks Atticus if telling the town that he killed Bob Ewell would be "sorta like shootin' a mockingbird." She realizes that he saved their lives, but would never be able to deal with being a hero in the spotlight. She then takes him home and as she stands on his porch, she sees everything through Boo's eyes. She sees the seasons fade from her 1st grade year to the present (3rd grade year).
Because we see the book through the eyes of Scout, her understanding of Boo and who he really is comes with her maturation. She has learned through her father that she doesn't truly know a person until she walks in his/her shoes. One of her last lines in the book was "Just standing on the Radley porch was enough." Through her understanding of Boo, her character truly grew from a young, superstitious child to a mature, young lady.
What you must understand is that Scout, the first person narrator, is sharing this story as an adult looking back on childhood memory. We really, can't know if her character developed as a child or if it is simply adult clarity. However, her initial portrayal of Boo Radley is that of a scary specter of childhood stories. All they know of Boo is the rumors that they must have heard and the details that their imaginations have fabricated. As the story progresses she recognizes his gentler side as he fixes Jem's torn pants, begins leaving them presents, covers them with blankets the night of the fire, and saves Jem from Bob Ewell. By the end of the novel, she recognizes how the community and they themselves had misjudged Boo. She knows he is a "mockingbird", and in addition she makes the connection between Boo and to the gray ghost she'd been reading about who had been innocent of everything he'd been accused of. This shows a cognitive and emotionally development, as well as a greater empathy for humanity.