Scout believes that only a woman can explain the social order to the children, because it is not important to Atticus. They have never had a mother to explain to them how important it is.
Scout is confused and frustrated by what her father is trying to tell her about the Finch family’s place in society. He is trying to explain these things to them because the trial has put a spotlight on them all, and because his sister Alexandra has asked him to explain it to his children.
The trial is putting enormous stress on him, and he shows it by asking Scout to stop playing with the comb, right after the lecture he has given them on “gentle breeding.” The whole thing is too much for Scout, and she starts to cry.
This was not my father. My father never thought these thoughts. My father never spoke so. Aunt Alexandra had put him up to this, somehow. Through my tears I saw Jem standing in a similar pool of isolation, his head cocked to one side. (Ch. 13)
When he sees how upset Scout is, Atticus folds. Scout says that Jem is frustrated and confused by the conversation. Atticus is not the one who tells them to act differently than others. He has always been the one to tell them to treat people like people, and has never put on airs. So he tells them to forget everything he has told them, realizing he has lost them.
This is when Scout makes her observation.
I know now what he was trying to do, but Atticus was only a man. It takes a woman to do that kind of work. (Ch. 13)
Atticus cannot explain the social structure of Maycomb, their place in it, and the importance of this to Scout and Jem. They have never had a mother to do it. Women are the ones who place importance on this, not men. Therefore they have grown up with Atticus’s egalitarian notions rather than the women’s understanding of the social function of things.
When Alexandra enters their lives, Scout and Jem become much more aware of the way in which Maycomb works on an adult level, and their place in it. Although Scout was aware of Maycomb's social strata, she never really considered her own importance or responsibility until her father tried to explain it. It is part of growing up, and part of the impact the trial had on her and her family.