After finding the first gifts in the knothole of the Radley oak tree, Scout doesn't seem to have a clue from where they have come. Jem believes they may have been left by a student with the intention of picking them up on the way back from school. But when the Indian Head pennies appear, Jem sees it as a sign of good luck.
Before Jem went to his room, he looked for a long time at the Radley Place. He seemed to be thinking again.
Although they continued to play the Radley Game, Jem receives another clue from where the gifts came when Boo's brother cemented the knothole. When Atticus explained that the tree was healthy, Jem seemed to recognize that Boo's brother's intent was to stop the gifts from being given. Jem finally comes to the realization that Boo was trying to be their friend on the night of Miss Maudie's house fire. A blanket had been placed on Scout's shoulders, and Atticus hinted that it could only have come from Boo. Jem then began "pouring out our secrets left and right in total disregard of my safety if not for his own..." He told Atticus that Boo could have
"... cut my throat from ear to ear that night, but he tried to mend my pants instead... he ain't ever hurt us, Atticus--"
When Scout understood that it was Boo who had placed the blanket across her shoulders,
My stomach turned to water and I nearly threw up.
From that point on, the children stopped playing the Radley Game, and
... tormenting Boo Radley became passe.
I guess this depends where in the book you are referring to. At the outset, Jem and Scout view Boo as a circus attraction. He is a human being that is built up in the children's heads as the stuff of myth and horror. As the story progresses Jem begins to understand Boo through acts of kindness like the mending of Jem's trousers or placing blankets on Scouts shoulders. External events like the persecution of Tom Robinson give Jem a more compassionate view of Boo. Jem also takes his cue from Atticus. Jem struggles to maturity by watching his father and, unlike Scout, starts to take it upon himself to do the right thing (ex. follows father to jail on the night Tom is threatened).