Importantly here, Atticus realizes that it is out of character for him to describe his family heritage. He ends his speech on the family by telling the children, "I don't want you to remember it... Forget it." Please see the question below as a reference to a full description of the scene.
This is a great question. Chapter 13 is an odd chapter. Aunt Alexandra comes to stay with the family for a time. She wants to stay, because she wants to give a feminine touch, especially to the children. Moreover, she wants them to know that they are "Finches." What she means by this is that they should be proud of their lineage and live up to their names. The Finches have been there since what seems like the beginning of time. She takes great pride in this, and apparently others do as well.
When she does not get very far with the kids, she asks Atticus to have a word with the children, so that they would take pride in their heritage.
Atticus suddenly grew serious. In his lawyer’s voice, without a shade of inflection, he said: “Your aunt has asked me to try and impress upon you and Jean Louise that you are not from run-of-the-mill people, that you are the product of several generations’ gentle breeding—” Atticus paused, watching me locate an elusive redbug on my leg.
Now this speech is out of place for two reasons. First, Atticus is not like Alexandra. He is not an "arrogant" person who places so much emphasis on external propriety. Second, Atticus, as he defends Tom Robinson, knows that upbringing and longevity in a place do not make a person good. For example, Bob Ewell's family goes back in time also! So, for Atticus to have this talk with Jem and Scout is hypocritical and inconsistent with his personality and beliefs.