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.