Some of the features of The Awakening that make it a "local color" story include the setting, the diction, and the characters.
The setting is the time and place in which a story occurs. Regarding the time period, The Awakening was written in 1899, thus it was written in a time when women raised children, and ran households. Women had no sexual feelings of their own, they did not need to be free, women were supposed to be content with staying home. It demonstrates this time period of society as the oppressive antagonist. As a result, this was a book that opened the door to the feminine revolution.
Also part of the setting is the location. Edna Pontellier is an upper-middle class white woman who has married a man from the upper-crust Creoles of New Orleans. She, her husband, and her children vacation on the Grand Isle, which is off of the coast of Louisiana. Here Edna is amused by the Creole lifestyle, how it is so different from her own Kentucky bluegrass upbringing. The Creoles are French speaking since Louisiana was owned by France until 1803. They follow the traditional views of Southern hospitality and diplomacy.
Along with the French societal structure, much of the language in The Awakening is written in French, thus emphasizing the diction, or author's word choice. More often than not, Chopin utilizes French words or French statements to describe situations and behaviors that would otherwise be "foreign" to non-Creole Americans. These European ideas are usually pertaining to stronger feminism, art, or music, thus implying that Europe was more advanced in the feminine revolution.
Regarding "local color" and character, Leonce Pontellier is a strict Creole man, and even is described as so by the Doctor, explaining that he must be the "man" of his household and his wife must follow his orders. Edna meets Robert on the Grande Isle, away from the city, and he runs off to Mexico to avoid his feelings -- his character demonstrates what a decent Southern man would do if put into a compromising situation. Madame Reisz speaks her mind and understands the taboos of her society, thus making her a social outcast and viewed at as a pariah by the other women. Madame Ratignolle is the perfect Southern wife -- one who "stands by her man" and adoringly coos over her children.