Scout ended up with a blanket on her shoulders because Boo (Nathan) Radley put it there when sneaking up on her. "'Someday maybe, Scout can thank him for covering her up.' 'Thank who?' I asked.' 'Boo Radley.' (96 Lee) Harper Lee added this part of the book in to provide more details about Boo and to get the reader more intruded for when Boo Radley will finally show up.
Throughout the book, Scout, Jem, and sometimes Dill have been constantly looking for this mysterious person, Nathan "Boo" Radley. He was accused of murder but never went to jail, however the 3 children don't think he is a very bad person. Boo has been leaving the children bread crumbs, like putting gifts in the tree hole, and fixing Jem's pants, but putting a blanket on Scout when she wasn't paying attention was the closest clue the kids have found to finding and proving Boo's innocence. Atticus Finch, Scout and Jem's father, always had a spot in his heart for the Radley's, but the rest of the town disliked them. The Radley's were known to be gloomy and crazy, but Atticus didn't think so.
This is similar to the book "Of Mice and Men" when everyone thought Lennie was a scary monster, but he was just a kind person with some mental issues. I don't think Boo necessarily has mental issues, however I do think he is perceived badly but is a good person.
Do you think that Jem and Scout will ever see Boo face to face?