Jem refuses to obey when Atticus tells him to go home. Atticus obviously is mindful of his children's safety at this point; he knows that the men have come to try and lynch Tom Robinson, and he does not want to risk his children being caught up in any violence. But Jem stoutly refuses to obey.
Jem does this because he (unlike the younger Scout) is also aware that violence might occur, that the men might hurt Atticus as Atticus is trying to protect Tom and Jem wants to help his father. From Atticus's point of view, it would be a bigger help if Jem went home to safety and took his little sister with him, but Jem wants to stay by Atticus's side.
Jem's refusal shows that he is being to assert his own individuality and manliness. He is on the verge of adolescence and is starting to make his own judgments and decisions. He still respects his father greatly, and likely will always do so. In the present incident, he defies parental authority not because he wants to be disobedient, but because he feels it's the right thing for him to do. He knows that Atticus, left alone to fight Tom Robinson's corner against the mob, needs support and he is not afraid to give it to him. This incident shows that Jem is cast very much in his father's mould; he too stands his ground in defence of what is right, and he shows himself to be physically and morally courageous like his father. Scout realises how alike they are in this moment as she looks from one to the other.
When the danger is over, Scout wrongly assumes that Atticus will get Jem into trouble for his disobedience. Instead, on the way home 'Atticus reached out and massaged Jem's hair, his one gesture of affection' (chapter 15). This shows that Atticus is proud of his son for supporting him at such a tense and difficult time.