Jem and Scout begin to part company when he goes back to the Radley place for his pants.
Scout is worried about Jem. She thinks that by going back to the Radley house, he is taking his life into his hands. Nathan Radley is supposed to have a gun and not be afraid to shoot anyone who comes near.
Scout and Jem are a few years apart, and there is usually a difference between their understandings of each other. However, Scout notes that her “periods of bewilderment were short-lived” usually. Yet when Jem insists on going back to the Radley place because he does not want Atticus to know he was there, and that he lied about the pants, Scout doesn’t get it.
This was beyond me. "Please," I pleaded, "can'tcha just think about it for a minute- by yourself on that place…"It's not like he'd never speak to you again or somethin'... (ch 6)
When Scout says that this is where they began to part company, she is looking back on the incident as an adult. She realized that Jem was getting to think more like an adult, and she was still thinking like a child. Therefore, they were not going to understand each other anymore.
Ironically, Scout realizing that she is not mature is a sign of maturity. She is able to try to emphasize with Jem enough to realize that she does not understand him.