Boo Radley puts the blanket over Scout’s shoulders because he feels protective of her.
When Miss Maudie’s house catches fire, most of the town comes out to help her. Atticus tells Scout and Jem to get up, and stations them well away from the fire but close enough to watch. This means he puts them in front of the Radley house, right by the gate. After the fire, Atticus notices something odd about Scout.
I noticed Atticus looking at me, first with curiosity, then with sternness. “I thought I told you and Jem to stay put,” he said.
“Why, we did. We stayed—”
“Then whose blanket is that?”
“Yes ma’am, blanket. It isn’t ours.” (Ch. 8)
Scout and Jem have no idea where the blanket came from at first. They were so caught up in watching the men try to rescue Miss Maudie’s belongings at the fire that they didn’t even notice when the blanket appeared. Nathan Radley was helping at the fire. The blanket could not have come from him. It is clear from where they are standing, in front of the Radley gate, that it could only have come from one other source—Boo Radley.
Scout is baffled by this event. She has spent most of her young life afraid of Boo Radley, and here he was helping her. It is an unusual interaction to be sure. Scout doesn’t make the connection with the presents left in the tree.
Jem’s first concern is that Atticus will tell Nathan Radley about the blanket, and that will get Boo in trouble. Jem knows that Boo considers himself their friend, and that putting the blanket on Scout’s shoulders was just another friendly gesture. He did it quietly, because he did not want to be seen. He just wanted to protect Scout.
Although they are not allowed to interact with Boo either, Jem confesses to Atticus that they have had interactions. He wants Atticus to leave the matter alone.
“…Mr. Nathan put cement in that tree, Atticus, an‘ he did it to stop us findin’ things—he’s crazy, I reckon, like they say, but Atticus, I swear to God he ain’t ever harmed us, he ain’t ever hurt us, he coulda cut my throat from ear to ear that night but he tried to mend my pants instead…” (Ch. 8)
Atticus understands the situation. He assures Jem that he will not try to return the blanket. Boo Radley came out that night because, like every other person in Maycomb, he wanted to help. Too shy to actually take part in the fire-fighting methods, he decided to watch over Scout and Jem.
This is a turning point in the relationship with Boo Radley for Scout. There is a long line of evidence that Boo cares about them. He laughed when they tried to make him come out, he left presents in the tree, and he mended Jim’s pants. Boo is looking out for them. It’s a good thing too, because if he hadn’t been, he might not have been able to save them from Bob Ewell’s murderous rage on Halloween night.