What evidence shows that Jem is beginning to understand more than Scout about Boo Radley?
When Jem and Scout start finding things such as gum, pennies, and soaps in the knothole of the Radley tree, they first think that it is someone's hiding place. Scout is just happy to find interesting trinkets in a peculiar place, but it's Jem who realizes that the trinkets are actually gifts. Jem is also the one who thinks that they should write the gift giver a thank you note. Jem dithers back and forth about involving Atticus in the situation, and he doesn't seem to be telling Scout his whole theory, either. The following shows Jem understanding the situation more than Scout:
"He had been on the verge of telling me something all evening; his face would brighten and he would lean toward me, then he would change his mind. He changed it again. 'Oh, nothin'.' . . . I pushed a tablet and pencil under his nose.
'Okay, Dear Mister. . . '
'How do you now it's a man? I bet it's Miss Maudie—'. . .
'Ar-r, Miss Maudie can't chew gum—' Jem broke into a grin" (61).
The above passage shows that Jem isn't telling Scout that he believes Boo is behind the gifts. Scout's theory is that it's Miss Maudie—showing she hasn't suspected Boo in the least. Jem, on the other hand, slips and almost says "Arthur" (Ar-r), which proves that he believes the gift giver is Boo.
Jem does a lot of growing up in the book. At the beginning we see him, Scout, and Dill obsessed with their quest to get Boo to come outside. As Jem matures more, we see that he is starting to see Boo as a person, not just a myth they have been told about. At the end of chapter 6, Jem has found his pants roughly stitched up, lying over the fence for him. He begins to think Boo had a hand in doing this. When Jem and Scout start receiving gifts in the tree, Jem knows this is the way to communicate with the elusive Boo.
When Nathan Radley seals up the knot-hole, Jem knows that the innocent communication between them and Boo is now at an end. Jem cries when he realizes that he can no longer have that connection to Boo anymore. What Jem doesn't realize is just how important he and Scout have become to Boo. Nor does he realize what an important and lifesaving role Boo will play in their lives in the near future. Their own lives are going to be held in the hands of Boo and they will come to see just how important all three of them are to each other.
The best evidence of Jem understanding Boo Radley and his life occurs after Nathan Radley cements up the hole in the tree. This tree has become an important conduit between Boo and the kids, Jem and Scout. As more and more trinkets and treasures are left in this knothole, Jem begins to realize that someone might be leaving gifts for them, specifically. Readers begin to see that it is Boo, and Harper Lee slowly brings Jem to the same realization. It is evident that Jem understands this as he stands on the porch with tears running down his face while looking at Boo’s house. He understands that Nathan Radley has cemented that hole in the tree to prevent Boo from leaving any more presents for the kids.