What is the function of relating the Finch family history at the outset of To Kill a Mockingbird?
There are at least two good reasons: first, author Harper Lee wants to establish a strong historical background of the Finch family ancestry in her exposition during Chapter 1. She shows a connection between Finch's Landing and Maycomb, establishes that the Finches are among the oldest and most respected families in Maycomb, and illustrates the family's connection with the Radleys. Secondly, Harper introduces all of the major characters while employing the literary device of flashback to show that all of the events of the novel have happened in the past--
"When he was nearly thirteen, my brother Jem got his arm badly broken..." (Chapter 1)
Scout's narration will be primarily in retrospect from a more adult perspective; we find that Jem, Scout and an elderly Atticus are all still alive when the story is being retold.
"We were far too old to settle an argument with a fist-fight, so we consulted Atticus. He said we were both right." (Chapter 1)