When Atticus says this, he is talking to Uncle Jack about the upcoming trial of Tom Robinson, and expressing his fears of how the trial might impact Jem and Scout. He is worried that Scout will let her temper get the better of her, and that she will try to fight anyone who brings it up; he indirectly warns her against this and asks her to refrain (she is listening behind the door and he knows). Then, he says that he hopes that he can "get Jem and Scout through it without bitterness, and most of all, without catching Maycomb's usual disease. Why reasonable people go stark raving mad when anything involving a Negro comes up, is something I don't pretend to understand." So, the "usual disease" that Atticus is referring to is how people do horrible things because of racism. Take for example the mob that gathers outside the jailhouse; otherwise decent, law-abiding citizens are overrun by fear and racism, and it prompts them to violence and evil. This disease-racism infects their blood and makes them behave crazily, just like some real, actual physical diseases can. So Atticus hopes that Jem and Scout get through the trial without catching the infectious hatred that will be spewed, and that they will "come to [him] for answers instead of listening to the town."
The usual disease is the mindset that the others have set up about racism. Most of the people in Maycomb believe that it is okay to be racist and that Tom deserves to be in jail even though he is innocent. Atticus is worried that Scout will get angry and get into a fight with people who are racist and do not look at the facts.