The quote you have selected comes from Chapter Five and comes just after Atticus Finch has told off his children for the games that they are playing surrounding the story of Boo Radley. In particular, Atticus is shown to respond in this way when Jem tries to defend himself by clarifying what he did and what he didn't say as if he were a lawyer. It is this that causes the strange reaction that Scout notices. Of course, we need to remember how the point of view impacts this quote and the novel as a whole. The narrator of this brilliant tale is a young girl who is not necessary aware of the full implications of everything she sees and hears. When she sees Atticus responding with a "suspiciously firm" mouth, she only remarks on it. However, for us, as an adult audience, we recognise that Atticus is actually trying desperately not to laugh at his children and Jem's feeble efforts to defend himself.
Atticus is therefore shown to be a very good father and to have an excellent knowledge of how to bring up his children. He understands that often he needs to pretend to be serious even when he is personally amused by the antics of his children and would love to express that.