To Kill a Mocking Bird Theory
Arthur Radley is Jem's biological father and possibly Scout's. Okay, so what we know is that Jem and Scout's mother is dead and their father (Atticus) has been raising them for a long time. We also know that weird objects have been showing up in the hollowed tree and that Arthur (Boo) Radley stabbed his father. What we don't know is why their mother is dead, why the weird objects have been showing up in the hollowed tree, and why Arthur Radley stabbed his father.
The explanation is, Arthur is Jem's real father. This is possible because Atticus' wife is dead, and if we use some creativity we can pretend that Arthur was in love with her. Let's say they had an affair while she was married to Atticus and she promises to divorce him and live with Arthur after Jem is born. But, she lied to him and says that she can't be with him. But they have another affair eventually and soon after Scout is born. But Scout is actually Atticus' daughter, and when she tells Arthur this he is furious. (Of course, she could have lied, so Arthur could be both Jem and Scout's real father) In a fit of rage, when she is walking home one night, Arthur Radley kills her and hides away in his house, leaving her corpse out on the street. Atticus looks at all the evidence but can't find an explanation (he is a novice lawyer at the time).
So, eventually Arthur confesses to his father that he had an affair with Atticus' wife and killed her. His father is freaked out and leaves to tell the police. So Arthur stabs him with some scissors and threatens to take his life if he ever tells what actually happened, so the fake story is a coverup. He is in denial and believes that Scout is also his daughter, which could be true if Atticus' wife lied to him.
Later on, Arthur knows he can't come out and tell everyone that he is their father, but he feels like he should do something for them. He leaves the gifts in the hollowed out tree for Jem and Scout to find, and after he does he continues to leave more gifts. But Arthur's father finds out and he seals up the tree to prevent Arthur from getting too attached.
Also, when Jem loses his pants, Arthur attempts to stitch them up because he wants to help his son in atleast some aspect. But he isn't very good at it and does a crooked job, but Jem isn't very good either (Jem says that when he got his jeans they were just how he would have sewed them up, so he gets this trait from his dad, Arthur)
Eventually, we get to the part where Scout is outside watching the fire burn. Arthur still has a shred of hope that he is actually her father and wraps her in a blanket and quickly slips away.
Some more assumptions is that Jem has a bad temper, just like Arthur did when he stabbed his dad. This explains why he throws the candy from the old lady into the fire in the book, even though it was a gift. And why he is generally mean throughout the book when Atticus is a gentlemen.
Then, skipping ahead throughout the storyline of the book, Arthur decides to attend the play (or whatever is going on at the school) to see Scout. Afterwards, he finds Jem and Scout getting attacked and he saves Jem's life. At the end, he reveals himself to Scout. He then strokes his son's face, knowing he may never get the chance to see him again. But he knows that he gave him the best gift of all: life. He proceeds to walk home with who he hopes is his daughter (Scout), but he eventually gives up on telling them because he realizes what a good father Atticus is
OR
Atticus isn't their father but still had an affair with Atticus' wife and he thinks that they are his children even though they aren't. He then continues to kill Atticus' wife and the rest is the same only without the lying and uncertainty
