"There is no king who has not had a slave among his ancestors, and no slave who has not had a king among his." What does this quote refers to?

This quote appears very early in the memoir as part of Keller's reflection on her ancestors. Before writing the sentence quoted, she notes that on her father's side she has a Swiss ancestor who was the first teacher of the deaf in Zurich and wrote a book on the subject. Keller refers to this as a "singular coincidence," then states:

it is true that there is no king who has not had a slave among his ancestors, and no slave who has not had a king among his.

By making this declaration, she dismisses the idea of putting any special weight on her ancestry or bloodlines. What she means is that who your ancestors are is unimportant: it is what you do that counts. She may have had one ancestor who made an important contribution, but she is not going to make too much of that. After all, everyone has had important and unimportant relatives.
This statement, so easy to brush over, is important, given the privileged Southern context in which Keller was born. Southern families such as hers tended to attach outsized importance to their ancestors and their blood lines, asserting that this ancestry made them superior to others. In the broadest terms, blood lines in the form of racial categories determined who did and didn't receives privileges in the South. By dismissing ancestry, Keller was making a statement about equality. She was a socialist and therefore rejected the idea of one social class as inherently superior to others.