In "The Story of an Hour," exposition provides us with the characters, the setting, the narrative frame, but its main purpose is to lead the reader away from the evolution in Mrs. Mallard. When we read, for example, that
Knowing that Mrs. Mallard was afflicted with heart trouble, great care was taken to break to her as gently as possible the news of her husband's death.
The exposition and the objective narrator help create expectations on our part that we will read about a tragedy in the life of a woman whose health is precarious.
The initial paragraphs, all of which are exposition, give the reader no idea that Mrs. Mallard's response to her husband's (purported) death is going to be anything other than conventional. In fact, her conventional response to the news is to
[weep] at once, with sudden wild abandonment, in her sister's arms.
At this point, the exposition--like a video camera-tells us that we are looking at a convention woman reacting to a terrible, but understandable, accident that kills her husband. And her response, which is to cry "with sudden wild abandonment," is a reasonable reaction to the loss of her husband.
When we get to paragraph 9, however, the third-person objective narrator, who has been recounting observable events, subtly becomes omniscient:
There was something coming to her and she was waiting for it, fearfully. What was it? She did not know.
Chopin has very skilfully altered the objective narrator and exposition to an omniscient narrator who can see into Mrs. Mallard's mind and heart.
The exposition, then, is designed to create expectations of normalcy that are shattered when the point-of-view shifts to a different type of narrator. From a practical standpoint, exposition is the easiest way for a writer to create the base line understanding of setting, characters, and story line in a story or novel. In Chopin's, though, exposition also creates expectations for the reader that will prove to be wrong.
Exposition is simply the part of the story in the beginning that introduces important background information.
The exposition of The Story of an Hour by Kate Chopin is in the very first sentence. Louise Mallard is a married woman with heart trouble.