This quote/KKK reference happens during Chapter 15, or what I like to call the "mob scene" chapter. There are two sort of "mobs" during this chapter. This first is a group of concerned friends and neighbors who visit Atticus to collectively warn him about the danger in following through on representing Tom Robinson (which likely relates to the social danger or damage to Atticus's reputation). As Jem and Scout watch Atticus talk to the men, Jem thinks the mob has come to hurt Atticus, but Scout soon realizes that there is no threat as these are men the Finches interact with daily.
This sets up a second "mob" scene, which happens when a group of armed men arrive at the jail, seemingly to lynch Tom Robinson in order to serve vigilante justice (in their mind, he is guilty even before the trial begins because he is black). Atticus predicted this possibility and stands between the mob and Tom in order to prevent the attack. However, it is the emergence of Scout (who thinks this group of men is safe because of her and Jem's previous misinterpretation of the first mob) which truly breaks up the lynch mob. Unlike the first mob, the second is made up of drunk strangers from the outskirts of town who truly intend violence. When Scout recognizes one of the mob members as the father of one of her classmates, she reminds the men of their individual humanity. Unable to harm a fellow dad in front of his children, the mob disperses.
To explain the significance, Atticus offers some insight on mob mentality, reminding the children that mobs are always made up of people. Between the two mob scenes, Jem expressed his concern that the first mob was a KKK group who had come to harm Atticus, and Atticus responds:
"Way back about nineteen-twenty there was a Klan, but it was a political organization more than anything. Besides, they couldn't find anybody to scare. They paraded by Mr. Sam Levy's house one night, but Sam just stood on his porch and told 'em things had come to a pretty pass, he'd sold 'em the very sheets on their backs. Sam made 'em so ashamed of themselves they went away." (Lee 196)
While the initial context of this scene suggests that Atticus is trying to comfort Jem and to tell him that he was worried over nothing, the broader context of the chapter and the two mobs reveals a deeper significance.
First, the statement is historically inaccurate. There was a KKK in 1920's/1930's Alabama (just as there still is a Klan today); however, Atticus is suggesting that the evil of these types of groups is a collective evil, and that the collection is made up of individual members who might otherwise be good and productive members of society. While it's easy to label the KKK as "bad", it's harder to label an individual community member and parent as patently bad.
So, as Harper Lee does so often in this book in this chapter and related to Atticus's KKK quote, she's suggesting the issue is not black and white. What's far scarier than a polarized, racist group like the KKK committing violence and fear tactics against upstanding Alabama citizens is the idea that the institutional racism of the time period was strong enough that normally "good" people would participate in subtle racism (like the first mob) or more significant violent racism (like the second mob). It would be too easy to have the KKK be one (or both of these mobs) and much easier to dismiss as a fringe hate group. It's much more interesting (and scary) to have those mobs made up of everyday Maycomb County friends and neighbors.