Why do Jem and Scout refuse to obey Atticus' command to leave him alone in front of the jail in To Kill a Mockingbird?

This happens in Chapter 15.  We are not exactly told why the two kids will not obey and go home, but I think we can make a pretty good guess.

I believe that the kids feel a great deal of loyalty to their father.  He has raised them alone and they respect him a great deal.  In Jem's case, he is probably feeling that he is becoming a man and should act like a man.

For both of these reasons, the kids would not want to abandon their father when he is in danger.  That is why they stay with him, in my view.