I believe that Scout, as the narrative voice for this piece, needs to give us the history of her family in order to place us in the context of Maycomb. We are entering a world that is unfamiliar to many of us due to time period as well as the small-town nature Maycomb is a place where your family's "history" is important to who you are in the town - the idea of coming from a "good" family is important to the people of the south at this time. The family that you are born into determines what is expected of you in life. This is important in two ways - it gives us a feel for what is expected of Scout (who then defies expectations on a routine basis - particularly in her refusal to become more of a "girl" and her preference for independence and independent thought) and it gives us expectations for Atticus (who also goes against them in defending Tom Robinson). We are establishing from the start a town in which stereotypes will rule, and then we are also establishing that it IS possible to break those thought patterns.
In my opinion, the reason for this is to give us a good idea as to how the Finch family fits in with the rest of the society in Maycomb.
As we see later in the book, the structure of the society in Maycomb is very important. Various kinds of people are expected to take on very different roles in the society. People like the Ewells are expected to be the scum of the society, people like the Finches are expected to be the cream (sort of how Aunt Alexandra acts).
By telling us who the Finches are, the author gives us a sense of where they fit in. At the same time, she gives us an understanding of how Atticus is something of a rebel because he does not take his expected role in society -- he doesn't act like part of the elite.
I believe that Scout tells the reader about the Finch family history in the beginning of the book, in order to help the reader to get an understanding of how the Finch family is deeply rooted in the south. Despite their southern connection Atticus is somewhat unique in the way he thinks and respects black people.
It was also of utmost importance to southerners during Scout's time to be able to know and share their family history. It was sort of like talking about being descended from a line of royalty.