When you read about the Jazz Age in a history book, you'll no doubt read about Prohibition, music, flappers, automobiles, lavish lifestyles, etc. All of these are included in the novel. Fitzgerald and his wife Zelda lived this lavish lifestyle themselves, so he was well aware of all of the things that made the Jazz Age unique. Gatsby's parties, the fashions of Daisy, the description of the cars, and the overabundance of illegal alcohol all come together to make it the "quintessential" novel of the Jazz Age.
The "Jazz Age" was a time of economic prosperity and moral decline. The term "quintessential novel of the 'Jazz Age'" is showing that The Great Gatsby is the most perfect embodiment of this specific time. This time period was characterized by post-WWI decadence and hedonism, as well as the growth of individualism. We see these things repeatedly throughout the novel through characters such as Daisy and Gatsby.