The Awakening is a great example of a story of a particular time and place. A primary conflict of the story is Edna's feeling like an outsider in Creole New Orleans and what she does to resolve that feeling. This society is different and unique, and Edna either needs to find a way to fit in or she needs to leave. The descriptions of clothing, homes, social obligations, attitudes about mothering and marriage, relationships between people are all shaped by this particular society. Edna has both small and large rebellions -- she is tan, she is aloof from her children, she doesn't open her home for visitors on Tuesdays, and she spends time with and appreciates Mde. Reisz. The local color atmosphere of New Orleans is created through descriptions of people, places, language, actions, and attitudes. The novel would be very different if the setting where different.
Three main related factors make Chopin's great short novel a local color story: the amount of time she spent describing things, the setting, and the language used. Regarding the time spent describing things, there is far more time spent on description than was needed for either plot or character development. This slows the action down—but it gives readers in a time before television a glimpse of another world. The setting is exotic—we have parrots, cottages, and "bridges" to them, all in the first page or so. And the language used includes everything from local references (the palmleaf fan) to literally different languages, such as the French the parrot speaks on the first page.
When other colored people are introduced in the novel, not much time is spent describing them or giving them a background. The reader can only truly relate to the whites in this book.