It's a scene from Captain America: The First Avenger. In this scene, he is talking to his friend who has always fought against bullies and never run away from them even though he was weak.
But for that meaning, isn't it right to say "He's too dumb to run away from a fight"?
He's too dumb not to run away. = Smart people would not run away but he runs away because he's too dumb.
He's too dumb not to run away. = He's too dumb to fight back. = He's so dumb that he can't fight back.