What does the blanket symbolize?

When Scout comes home from the fire at Miss Maudie's, Atticus sternly asks her where she got the brown blanket that is thrown around her shoulders. It is not one of theirs. Scout is entirely bemused: she has no idea. Finally, Atticus relents and tells her it was

Boo Radley. You were so busy looking at the fire you didn’t know it when he put the blanket around you.

The blanket symbolizes the quiet but active way Boo is looking out for Jem and Scout. He has taken an interest in the two.
Scout, however, is shaken when she finds out that Boo was close enough to her to drop a blanket around her shoulders. To her, he is still the frightening bogeyman.
Boo's gesture is another indication that all is not as it seems on the surface. Scout has to learn to discern between the gossip that circulates in Maycomb and the reality of how people act. The blanket represents the thoughtfulness that characterizes Boo. He is a guardian angel for Scout and Jem, rather than a bogeyman.

It's a cold winter's night in Maycomb, and Miss Maudie's house is on fire. Scout is watching the flames, standing on the sidewalk, shivering outside the Radleys' front gate. Seemingly from out of nowhere, someone places a nice warm blanket on her shoulders. But she's too engrossed by the fire to notice. It's only when she returns home later that evening that she realizes there was a blanket on her shoulders all that time and that it had been put there by Boo Radley.
Slowly but surely, Boo has been reaching out to the Finch children. We saw this earlier when he left them some personal items in the knothole of a tree. Boo's blanket symbolizes his growing protectiveness towards Scout and Jem and foreshadows events later on in the story, when he'll protect them from Bob Ewell.