Excellent question. The quotes that introduce each chapter take the form of non-fiction quotes relating to the scientific study of bees and their lives. Bees are of course an ever-present reality in the novel, but what is interesting is how the quotes relate bees and their activity to the lives of the characters. For example, the quote introducing Chapter Eight refers to the necessity of bees having companionship and how honeybees "soon die" if they are isolated from other bees and lack companionship and support. This chapter includes Lily's first introduction to the bees and the way that August tells her to "send them love."
The resulting swarm of bees that crowd around Lily triggers a kind of spiritual reverie. As she reflects back on it, note how Lily thinks about what happens:
I knew that these bees were not a plague at all. It felt like the queen's attendants were out her in a frenzy of love, caressing me in a thousand places. Look whose here, it's Lily. She is so weary and lost. Come on, bee sisters. I was the stamen in the middle of a twirling flower. The center of all their comforting.
Lily, in spite of her attempts to run away from companionship, is forced to realise that she needs the same companionship and support that bees themselves need, and finds it in this chapter through a mystical union with the bees, just as she is finding it in a more practical nature through her friendship with August and the love that August shows her. Thus we can see from this example that the quotes that begin each chapter implicitly relate to the action of that chapter and the characters in it.