This quote, at the very end of Chapter 13, follows a very awkward conversation between Atticus and Scout. Aunt Alexandra is in town, and basically impresses upon her younger brother to talk to Scout (in particular) about being a lady. He never actually uses these words - but as he attempts to quote Aunt Alexandra it is clear that this is what she means.
Scout herself has no idea what Atticus is talking about. He uses Aunt Alexandra's words, "Not run-of-the-mill people," and "product of several generations' gentle breeding." It is clear Atticus himself doesn't buy into it, and cannot sell the idea to Scout that she should be acting lady-like, wearing dresses, learning manners, preparing to be a debutant one day, etc.
In fact, Scout doesn't understand it until much later (as she tells the story from an adult perspective) that it was impossible for Atticus to guide Scout to becoming a young woman "of society." It was really a mother's job and it would take such a mother to role model it first and teach it that way.
Scout ends Chapter 13 with this quote that reflects her thoughts about Atticus trying to implement Aunt Alexandra's decree that the children begin acting more like Finches. Atticus has attempted to explain Aunt Alexandra's ideas about the importance of the family and its "gentle breeding." Jem and Scout had already heard most of the family history from Atticus, and they knew the true facts, so they weren't buying into their aunt's beliefs. After his sister gave him an ear full, Atticus attempted to convey Alexandra's desire that Jem and Scout begin acting more gentlemanly and lady-like. When Scout began to cry, Atticus gave up the attempt, and Scout knew "he had come back to us." It was a futile attempt by Atticus to take a feminine approach to parenting his children--one that they all knew was unlike his typical style--and Scout recognized that "it takes a woman to do that kind of work."