Scout is growing up by trying to climb into Jem’s skin and walk around in it because it shows that she is mature enough to think about other people.
As much as it is a novel about racism in the deep south, this is a novel about a little girl’s coming of age story. When the story begins, Scout is a hot-headed kid who thinks mostly about herself and does not really understand how the world works. As time goes on, she begins to care more about others. Atticus tells her that if she tries to look at things from another person’s perspective, she’ll understand people better.
When Jem leaves his house in the middle of the night to retrieve the pants he lost at the Radley house, Scout simply cannot understand why he would go. She thinks he will be killed, either because Boo Radley is dangerous or because Nathan Radley will shoot him. So she tries to put herself in Jim’s skin as Atticus advised.
[If] I had gone alone to the Radley Place at two in the morning, my funeral would have been held the next afternoon. So I left Jem alone and tried not to bother him. (Ch. 7)
Scout is unsuccessful. She is still too young to understand that it is more important to Jem that Atticus trust him think that he listened, so that he can see him as an adult, than to escape punishment for disobeying. Yet the fact that she tried to understand what Jem did, and put herself in his place, shows that Scout is taking Atticus’s advice and learning empathy.
Scout’s development of the skill of empathy, a key skill of adult life that many of the full grown adults in Maycomb have yet to master, is one of the key journeys of the book. She demonstrates empathy for Mayella Ewell, and for Tom Robinson, and later for Boo Radley. As Scout matures, she learns the lessons that we all need to learn at one time or another—that when you look at something from another person’s point of view, you often change your own.
By Scout saying she is walking inside of Jem's skin and crawling around it, basically means she is walking in his shoes. She can finally understand Jem, because she is seeing things through his perspective. Earlier in the novel, Scout tried to walk in Walter Cunningham's, but couldn't because she was too young to understand. By the end of the novel, she has matured a lot and which is why she says that she can now walk in other's shoes because she is now mature.