This is a good question. There are several places in the book that show that Jem and Scout grew through this experience - a coming of age, if you will.
First, from a general point of view, they grew because they watched a very important trial featuring adult subjects such as rape, physical abuse, and racism. Not only did they witness the trial, they also experienced the ramifications of it as they were persecuted to some degree because Atticus was defending Tom.
Second, Jem and Scout had to face the fact that Atticus lost, and Tom died trying to escape. This tragedy made Jem in particular grow up. The happy thoughts of youth were now gone, as he experienced injustice in the form of racism.
Third, at the end of the novel, evil almost killed both Jem and Scout. Through this they experienced how wrong the world can be. And through it, they still understood that they must continue to do what is good and noble. The conversation between Scout and Atticus at the end of the book shows that Scout matured greatly. She was now able to walk in the shoes of another.
Atticus sat looking at the floor for a long time. Finally he raised his head. “Scout,” he said, “Mr. Ewell fell on his knife. Can you possibly understand?”
Atticus looked like he needed cheering up. I ran to him and hugged him and kissed him with all my might. “Yes sir, I understand,” I reassured him. “Mr. Tate was right.”
Atticus disengaged himself and looked at me. “What do you mean?” “Well, it’d be sort of like shootin‘ a mockingbird, wouldn’t it?”