The mockingbird is one that sings beautifully, but never hurts anything; hence the saying that it is a sin to kill one.
Boo Radley is mockingbird-like in that he never hurts anyone, but only tries to be helpful. He had some issues in his past, but most of those can be traced to his parents and his older brother. He is reclusive because he has been forced to be that way, and over the years it has become his way of life. But it isn't because he doesn't love other people, or because he doesn't want to be a part of the world.
Examples of Boo's kindness abound in the book, in the ways he treats the children. He leaves treats for them in the knothole of the tree, he puts a blanket around Scout's shoulders during the fire, and eventually he saves them from death at the hands of Bob Ewell. He loves "his children," and would do anything for them.
In the end of the novel, Atticus and Heck Tate discuss Boo and what to do about his involvement in the Ewell case. They decide to keep it a secret, because to expose Boo would be to drag him into the spotlight, where he would be very uncomfortable. Like killing a mockingbird, it would be a sin to do anything to hurt Boo Radley, after all the good he has done.