In "To Kill a Mockingbird" why does Jem say that they should not tell anyone at school about the incident?

In chapter 10, the incident you are referring to is when Atticus very deftly shoots the old, rabid dog Tim Johnson.  Jem had no idea that Atticus could shoot; in fact, Scout had just gotten done exlplaining how "Atticuse was feeble" and how at school, when other kids bragged about how their dads could do this or that, "there was nothing Jem or I could say about [Atticus]" to earn bragging rights about their dad.  So, in that sense, Jem was a bit ashamed of his dad.  He wished, like most little boys, that his dad could do something super cool so that he could brag about him at school.  So, when Atticus shoots down the old dog in one clean and very precise shot, Jem is stunned.  He tells Miss Maudie, "he never said anything about" the fact that he could shoot.  Miss Maudie explains that Atticus never bragged about it because he didn't want to make other people-who couldn't shoot-feel bad because of his talent.  Miss Maudie concludes by saying, "People in their right minds never take pride in their talents."

So, Jem doesn't want to tell anyone at school, because he is trying to follow Miss Maudie's wise statement, and to respect Atticus's wishes.  He says about Atticus, "if he'd wanted us to know it, he'da told us."  He is very proud of his father, but since Atticus doesn't go around bragging about it, Jem decides to respect his wishes.  He tells Scout not to tell anyone, because if Atticus wanted people knowing, he would have told them.  We see Jem making a rather mature decision to not brag about something that his father didn't want bragged about, and to respect the wishes of his elders.  I hope that helps a bit!  Good luck!