When someone is humble, it means that they are not selfish and they do not brag about their talents or abilities. Atticus demonstrates what it means to be humble during the mad dog incident, when he is appointed as Tom's lawyer, and after the trial with the black community.
First, during the mad dog incident in chapter 10, the sheriff tells Atticus to shoot the dog because he's the best shot in the county. Even though Atticus hasn't picked up a gun in decades, Sheriff Tate knows that Atticus is only being humble when he declines the honor. The sheriff warns Atticus that if he doesn't shoot now, the dog will go into the Radleys' house. Eventually, Atticus consents to take the shot, only because the sheriff pleads with him and because of inevitable danger. The shot rings true and the dog goes down without any suffering. Jem and Scout are amazed because they never knew their father had such a talent and was called, "One-Shot Finch" as Miss Maudie reveals (97).
Next, in chapter 16, Scout overhears some men at the court talking about Atticus being appointed by Judge Taylor to defend Tom Robinson. This is news to her because her father has always said that it is very important to him to defend Tom Robinson because it is the right thing to do. However, her father's humility on this one point could have proven valuable for her when Cecil Jacobs, Francis, and Mrs. Dubose accused him of being a "ni****-lover". Either way, Atticus is humble because he doesn't announce to everyone that he was appointed as Tom's lawyer; rather, he steps up and does his duty willingly.
Finally, the day after the trial of Tom Robinson, Atticus wakes up to loads of edible gifts from the black community who thanks him for defending Tom honestly and courageously. Atticus is amazed to receive such an abundance of food from Tom's family and friends that he sheds a few tears over it. Atticus demonstrates his humble nature because he doesn't demand anything from anyone for his efforts. He says to Calpurnia, "Tell them I'm very grateful . . . tell them they must never do this again. Times are too hard" (213). Atticus understands what sacrifices were made to give him so much food, and he doesn't want them to suffer for it.