The blanket that Boo Radley places on Scout's shoulders as she stands in front of the Radley yard symbolizes the gentle, caring nature of Boo and his relationship with the children with whom he has involved himself (from a distance).

When Miss Maudie's house becomes engulfed in flames, Atticus, who has awakened the children, sends Jem and Scout down the street so that they will be a safe distance from the dangerous flames. He instructs them to stand in front of the Radley Place and stay out of the way of everything and everyone involved with fighting the fire.

Certainly, the children become mesmerized with the raging fire and the desperate situation of the rotund Mr. Avery, who in his attempt to escape from an upstairs window by jumping onto a mattress he has thrown down, becomes wedged inside the window frame. Struck with fear that Mr. Avery may be burned, Scout hides her head under Jem's arm.

It may be at this point that Boo Radley places the blanket upon Scout, who is absorbed in the frightening moment. At any rate, she does not become aware of the brown woolen blanket secretly placed upon her shoulders until her father scolds her and Jem for moving from their designated position. After Jem explains that they have obeyed him and not moved, Jem realizes who must have put the blanket on them. Atticus, understanding, agrees that they shouldn't attempt to return it:

"We'd better keep this and the blanket to ourselves. Someday, maybe, Scout can thank him for covering her up."
"Thank who?" I asked.
"Boo Radley. You were so busy looking at the fire you didn't know it when he put the blanket around you."

This solicitous act of the reclusive Boo Radley is not only significant in and of itself; it also foreshadows Boo's heroic act for the children he has come to love that occurs in the final chapters.

Before this point in the story, Boo was known to all of the town, and the neighborhood children, as the "malevolent phantom," a bad character who was responsible for spying in people's windows, harming small animals, and stealing things.  Although none of these things are true of Boo, kids still are afraid of him, and he is a bit of a legend in the town.

As Scout stands outside in the cold, watching Miss Maudie's house burn to the ground, she realizes at some point that she was "clutching a brown woolen blanket" around her shoulders.  She doesn't know where the blanket came from.  It is confirmed by Atticus and Jem that Boo put it there.  She hadn't even noticed.

So, Boo, this supposed evil person, had noticed Scout that Scout was cold, and slipped a blanket around her to keep her warm.  This action symbolizes friendship, kindness, generosity, and, as is a strong theme in this book, how people aren't always what we think they are.  People aren't what they appear to be on the surface, and we shouldn't judge them before we get to know them.  Boo, on the surface a strange social recluse with a reputation for harm, was really a harmless, friendly, kind human being that noticed Scout's need and served her selflessly.  I hope that those thoughts help; good luck!