Chopin uses the storm as a foreshadowing of the coming stormy affair between Alcee and Calixta, and also, a symbol of Calixta's underlying unhappiness in her marriage. There is not much else to indicate that Calixta is unhappy or unsettled about her life. We get only descriptions of her sewing, but as she sewed, she "felt no uneasiness for their safety," referring to her husband and son. She is just busily sewing, without much thought in her head, so there is no indication that she is on the cusp of committing an affair with an old flame. Chopin has her declare of the rain, "My! what a rain! It's good two years sence it rain' like that," possibly symbolizing that it has been quite some time since she has felt passion or happiness in her life either. So, it is a symbol of her emotional state. So, the storm symoblizes her underlying unrest, and foreshadows the change that is about to occur.
Along with being a symbol of Calixta's unhappiness in marriage, and foreshadowing of the coming affiar, the storm also is a necessary element of the action, or the plot. Without the storm, Alcee would not have needed to seek shelter in Calixta's house. That event was necessary in order for the affair to occur. So, it is also just an elemental action or bit of plot that is necessary for the events of the story to occur as they do.
Chopin often uses the weather to symbolize events or feelings in her stories. In this story, the storm is the unrest and coming changes that occur. In another story of hers, "The Story of an Hour," Chopin uses a sunny sky and perfectly green leaves and good weather to symbolize the main character's happiness after her husband dies and she realizes that she is free. So, when reading Chopin, keep in mind that she uses the setting and the weather to model her main character's inner feelings quite often. I hope that those thoughts help; good luck!