This statement from Nick's opening remarks establishes 1) the theme of status/class in the novel and 2) aligns Nick with class concerns.
While other characters are depicted as clambering after wealth and status, often ostentatiously, to a far greater extent than Nick, it remains true that Nick is also concerned with his own status. Not long after this statement is made, Nick discusses his intentions to become an intellectual (a well-rounded man).
Naturally, Gatsby is the best example of concerns with status and class distinctions in the novel. However, as Nick's statement shows, Gatsby is just one of the many characters in the novel who anchor their identity with class identity.
First, let me say that I agree with all of the above interpretations. Yes, this statement introduces the theme of "status/class." Yes, it also shows Nick has "class concerns," but I would also add that it nicely plugs him in immediately to the "new rich" category instead of the "old rich" category.
Keep in mind that the people in the novel are very stuck on which "kind" of rich they are, so much so that they insist on living in different places. The old rich, the people who have had unlimited funds for as long as they and their families can remember, live in East Egg. They have no need to work and simply inherit their money from generation to generation. The new rich, the people who have newly acquired funds they have (generally) earned, live in West Egg. These people have either worked for their money, acquired it by less than appropriate means, or had some kind of fortuitous circumstance. People in West Egg are generally looked down upon by the people of East Egg.
The information above puts Nick's comment into context. Here, Nick is proud that his family has been "prominent" and "well-to-do" for "three generations." This establishes him immediately as the "new rich." He is only the third generation to have money. Period. East Eggers would immediately look down upon him for that. There is also a suggestion that, somehow, this money has been earned and not necessarily inherited. Again, this is a "new rich" kind of idea.
The reader, then, should not be surprised when Nick quietly slips into the neighborhood of West Egg and inhabits the "shack" right next to Gatsby's mansion.
Nick's statement is an important one in the beginning of the novel because it both introduces the overarching theme of class and status, but also hints at the more subtle theme of security in one's social class and theme.
Money, power, and status are prominent in almost every chapter of the Great Gatsby. Nick, who is from "prominent, well-to-do people," represents the sort of old money character that is immediately valued and successful in the novel. Consider how the Buchanans and Jordan are among the sort of higher class and stable positions in the novel, whilst Gatsby–who may comparatively have more wealth–is both unstable in his financial situations and must fight to preserve his social image.
Thinking about Gatsby and his image also makes Nick's quote interesting because it reminds of the old money/new money struggle, and how true social wealth more easily comes to those who have a history of being wealthy. Consider how Gatsby has struggled and plotted to create an image that will woo Daisy, yet this world is incredibly more precarious that that of his neighbor's.
Nick Carraway is a classic example of an unreliable narrator. In the opening lines of the book he claims to "reserve all judgement" and yet spends the rest of the book in almost detached categorization of the people that float around him. Some of F. Scott Fitzgerald's best lines come out in Carraway's brutal, unflinching judgments. Consider this line from our first introduction to Tom Buchanan:
"I felt that Tom would drift on forever seeking, a little wistfully, for the dramatic turbulence of some irrecoverable football game."
That does not sound like the carefully moderated tone of a man inclined to reserve all judgment.
One of the prominent themes in "The Great Gatsby" is class identity. Where do you fall in society? Do you live in East Egg or West Egg. Many of the characters in the novel (most of the prominent ones, in fact) seem to believe that your class determines almost everything about you, even your moral structure and your personal worth.
Carraway claims to be above all that. He reserves judgment, choosing to recognize that "all the people in this world haven’t had the advantages" Carraway has. And yet, not one page into the novel, he makes sure we know his class, where he comes from. Carraway proves that he falls just as much into the trap of class identity as the other characters in the novel and sets himself up as an unreliable narrator.
Although a calculated, underlying motive exists in Fitzgerald's usage of this background material, it must first be appreciated as establishing Carraway's character. Coming in the opening pages, the outline of Carraway's shape is being brought into focus, foreshadowing the themes, contrasts, and tensions of the novel.
Fitzgerald is at once using this statement to provide context to the similarities and differences between Nick and the yet-to-be introduced characters. The fact that the Carraway stock is "prominent," "well-to-do," and has been so for three generations establishes Nick as affluent, comfortable in his status, and entrenched in this identity. While these factors may endanger Nick as being no better than other self-seekers in the novel, Fitzgerald undoes much of this impression. Nick almost seems to laugh at the family by acknowledging the false belief of the Carraways descending from the Dukes of Buccleuch, clarifying that they in fact derived from a wholesale hardware man who shirked his duty in the Civil War. Further, his father taught Nick that he should be cautious of harshly judging others because of the great advantages of his own life. This balance would be unthinkable in characters such as Tom and Daisy, further bringing the contrasts between Nick and others into relief.
Nick's background is of great importance because it establishes him as similar to those he comes to disapprove of. Had Nick come from lowly stock, his disdain for the rich and selfish would have been both obvious and natural. Instead, the reader is allowed to see Nick's similarities before divergences become a factor. His personality takes on greater depth by showing how his background--and more importantly his choices--are relevant to the upcoming tensions in the novel.