Atticus wants to return the blanket, but Jem does not want Boo to get in trouble for putting the blanket on Scout’s shoulders.
Atticus wants the wrapping paper because he wants to return the blanket that Boo put on Scout's shoulders during the fire at Miss Maudie’s house. While everyone was watching the fire, Boo Radley snuck behind her and put a blanket on her. It was really just a kindly gesture. Atticus chuckles and decides he isn't concerned, but Jem does not want anyone in the Radley house to know that Boo was anywhere near them. They cannot return the blanket, and they can’t tell Nathan Radley.
"Atticus, no sir!"
Jem seemed to have lost his mind. He began pouring out our secrets right and left in total disregard for my safety if not for his own, omitting nothing, knot-hole, pants and all. (Ch. 8)
Scout is concerned because they have been keeping all of this a secret from Atticus too up until now. They are not supposed to be interacting with Boo Radley, because they are supposed to be leaving him alone. However, in this case, Boo Radley is the one interacting with them.
The children have spent all summer trying to get Boo Radley to come out. Now, Boo Radley finally did. Scout did not see him, and neither did anyone else, but the secret is out. He is their protector. He left them little gifts in the tree, and Nathan Radley got upset. This is why Jem is concerned that Boo will get into trouble if anyone finds out about the blanket.
"Mr. Radley, ah- did you put cement in that hole in that tree down yonder?"
"Yes," he said. "I filled it up."
"Why'd you do it, sir?"
"Tree's dying. You plug 'em with cement when they're sick. You ought to know that, Jem." (Ch. 7)
The part about the tree dying is a lie. There is nothing wrong with that tree. He just doesn't want Boo leaving anything in there for the kids to find again.
While it may seem strange that a grown man would want to befriend children, Boo Radley leads a lonely and isolated existence because of his childhood as a juvenile delinquent and the effects of overbearing parents, and now the overbearing Nathan Radley.
Atticus understands why Jem does not want anyone to know that Boo has a relationship with Scout and Jem, especially Nathan Radley. He realizes that the children mean Boo no harm, and Boo certainly means them no harm. Scout is also starting to realize that Boo is also not a scary man either. He is just a lonely one.