What does the quote at the start of chapter 3 in "The Secret Life of Bees" mean?

"New beekeepers are told that the way to find the elusive queen is by first locating her circle of attendants."

When looking into a hive the majority of bees are females.  In order to locate the queen one must do exactly what the quote states.  When you find a large group of bees surrounding a larger female bee that is probably the female. These are her associates.  There are few male bees, or drones, in the hive and they aren't very useful.  They are simply there to fertilize the queen and then it dies.  One fertilization is enough to supply the eggs for the queen's life time.  This quote, as are most of the quotes at the beginning of each chapter, is from the book "Queen Must Die and Other Affairs of Bees and Man"

In relationship to the book the quote means that if you look at people from the point of view of who they associate with, you can quickly tell who is the "queen" of the hive.  Many things in the lives of bees and people are secret, but we can discover a lot by who their associates are.

What is the significance of the quote at the beginning of chapter 3 in The Secret Life of Bees?

New beekeepers are told that the way to find the elusive queen is by first locating her circle of attendants.

While some think of the queen bee as being August, another possible interpretation is that the queen is Lily’s mother. Lily on her quest to understand herself, is searching for her mother throughout the story. Her mother is dead, so she has to look for the ghost of her mother in places where people would have known her.

The “circle of attendants” is a metaphor for those who were around her mother, the people that she spent time with and would have information about her. Finding out more about her mother is the reason why Lily is searching out the honey maker that put out the image of the black Madonna.

It is clear from her questions in chapter three that Lily is looking for information, to get to know her mother, from August Boatwright. She says, “Don’t you think my mother must’ve known her? . . . It couldn’t be just a coincidence”—showing that she is searching for her mother, and the place she has to go for info is to those who might have known her and her reasons for leaving.

Lily gets her best clue by finding the honey at the store in Tiburon. The other patrons of the store are able to point her to the Boatwright house and Lily goes there to discover who her mother was.

The entire book centers on Lily coming to terms with who she is in the face of her abusive and lonesome upbringing, and the search for her mother is what helps to bring her closure.

Your excellent question of course relates to the way in which the quotes that begin each chapter relate to the contents of the chapter. What is interesting about Chapter Three is the way that we see Lily making an important discovery that will lead her to the queen bee, August, who will have such an important impact on her life. Note what leads her to discovering August and her whereabouts is seeing the picture of the black Virgin Mary in the shop and asking the shop attendant about it. Of course, finding one of the queen bee's attendants, in the form of a shop that sells August's honey, allows Lily to locate the queen bee herself, as the shop attendant gives her the directions to August's house:

Oh, sure, it's the darndest house you ever saw. Painted like Pepto-Bismol. Your grandmother surely's seen it--you go through town on Main Street until it turns into the highway to Florence.

Thus it is that we see Lily making another important step in her journey to self-discovery.

In The Secret Life of Bees, what is the significance of the chapter 7 quotation?

Chapter 7 starts with the following quote:

How did bees ever become equated with sex? They do not live a riotous sex life themselves. The hive suggests cloister more than bordello. (The Queen Must Die: And Other Affairs of Bees and Men)

Essentially, this quote is saying that, though people often equate bees with active sex lives akin to prostitutes, they are actually much more like nuns who live and work together and abstain from sex. The comparison to nuns is also indicative of the fact that Lily has been taken in by a group of sisters who all worship the Virgin Mary.

This quote is significant because it foreshadows Lily's sexual maturation and the fact that she will soon begin to develop romantic feelings towards Zach. People often assume that teenagers, like bees, have "riotous sex lives," but that isn't the case for Zach and Lily. Like worker bees, they spend their time working alongside each other. While they work, Zach and Lily share details of their respective lives and eventually develop a close friendship.

As the chapter progresses, we see Lily discover that she has feelings for Zach, and an innocent romance begins to bloom. However, it isn't typical of what one might assume of two teenagers who are able to spend a lot of time together. Their physical contact is limited, and though they come close when they are tending to some hives, they don't even kiss.

Zach later reveals to Lily that he does have feelings for her but can't act on them because she is white. This adds another layer of importance to our understanding of the quote at the beginning of the chapter; like nuns who choose to be celibate, Zach and Lily have to choose to overlook their romantic feelings for one another due to the racist climate they live in.

The quote I believe you are referring to is, "How did bees ever become equated with sex? They do not live a riotous sex life themselves. The hive suggests cloister more than bordello." It refers to the talk about the "birds and the bees" that every parent dreads having with their child sooner or later. Lily and Zach are becoming more and more interested in each other as those of the opposite sex do. Bees mate only with the queen, and most drones do not mate at all. How bees came to be in the phrase above is funny given their mating habits (or lack of them). Each chapter begins with a quote from a nonfiction book about the life of bees. The quote, in some way, relates or foreshadows the human events in each chapter.