Boo shows that he wants to become friends by putting the blanket on Scout's shoulders.
Boo Radley wants to become friends with the children because he is lonely, and they have shown an interest in him.
We know that Boo Radley is reclusive, and rarely comes out of his house. However, from his house, he sees the children playing all summer. They often re-enact scenes from his life, and come up with a game to try to make him come out. They even go so far as to try to leave him a note. They are friendly toward him, when most people ignore him or have forgotten about him because of his misspent youth.
Jem said placidly, "We are going to give a note to Boo Radley."…
Jem was merely going to put the note on the end of a fishing pole and stick it through the shutters. If anyone came along, Dill would ring the bell. (Ch. 5)
This kind of behavior must have been seen as endearing and cute by Arthur Radley and not rude or mean. The adults thought that the children were bothering him, but he was lonely and just happy that someone was finally paying attention to him. He left the children presents in the form of pennies, gum, soap dolls, and a watch.
When Jem leaves his pants at the Radleys, he is concerned because he thinks he will get in trouble. He goes back to get them, and he finds someone (Boo) has left them for him and sewn them up for him.
Boo Radley starts to take a more active interest in the children though. In Chapter 8, Scout finds out for sure who their secret benefactor is during the fire. When Jem sees the blanket, he immediately is worried about Boo. He does not want him to get in trouble for being involved with them since Atticus knows now that he has been acting friendly toward them. He finds that Atticus is sympathetic to Boo though.
Scout is afraid when she first realizes that Boo Radley was so close to her.
You were so busy looking at the fire you didn't know it when he put the blanket around you."
My stomach turned to water and I nearly threw up when Jem held out the blanket and crept toward me. "He sneaked out of the house- turn 'round- sneaked up, an' went like this!" (Ch. 8)
Jem tells her that if she had turned around she would have seen him. He has a point. Boo came close enough to Scout to put a blanket around her shoulders, and was worried about her enough to do it, but still did not let himself be seen. He is still shy, and just wants to remain in the shadows. He wants to make entrances in their lives, but not be a part of their lives. He is taking small steps.
Boo Radley is an adult, but he has some childlike qualities. This is why he is interested in being friends with Scout and Jem. He has been trapped in his house for a long time. The children are the only ones who have been nice to him. In this chapter, he makes an overture of friendship and Scout realizes that Boo Radley really is not all that scary after all. It is one of the first signs that she is growing up.