What is Atticus trying to do at the end of Chapter 13 of To Kill a Mockingbird? Why does Scout say that this is better left to a woman?

Atticus is trying to explain to Scout and Jem about the importance of the Finch name, but that (ancestry, heritage) is Alexandra's passion.

Aunt Alexandra is the one who cares about heredity.  To Atticus, people are people and all people deserve respect.  However, he agrees to talk to his children about the “facts of life” so that they will understand that they are Finches, and the importance of being a Finch.

Atticus tries to explain...

Atticus is trying to explain to Scout and Jem about the importance of the Finch name, but that (ancestry, heritage) is Alexandra's passion.

Aunt Alexandra is the one who cares about heredity.  To Atticus, people are people and all people deserve respect.  However, he agrees to talk to his children about the “facts of life” so that they will understand that they are Finches, and the importance of being a Finch.

Atticus tries to explain the concept of “gentle breeding,” which basically means that wealthy families marry other wealthy families.  Since the Finches are the aristocratic class of Maycomb, that means that Scout and Jem should try to “live up to” their name and act like Finches.

“She asked me to tell you you must try to behave like the little lady and gentleman that you are. She wants to talk to you about the family and what it’s meant to Maycomb County through the years, so you’ll have some idea of who you are, so you might be moved to behave accordingly”… (Ch. 13)

The concept of being a Finch means nothing to Scout and Jem.  Scout would rather run around in overalls than wear pearls.  Jem, who is older, does understand what Atticus is trying to say, but it contradicts everything they have been raised to believe.  Scout is confused by the concept of some people being better than others just because they have been in Maycomb longer.

Somewhere, I had received the impression that Fine Folks were people who did the best they could with the sense they had, but Aunt Alexandra was of the opinion, obliquely expressed, that the longer a family had been squatting on one patch of land the finer it was. (Ch. 13)

The Ewells have been on their land a long time, but they are not a "fine family."  It is confusing to Scout, and Atticus is not able to explain.  Scout’s comment that it takes a woman to explain this seems to indicate that she feels that women care more about heredity than men.  That is the case in her family.  Scout knows that Atticus doesn’t really care about being a Finch.  He acts according to his own conscience, even at the risk of his reputation.