The Awakening is considered an example of literary realism because of its use of local color (already covered in detail by the other answers) and the ordinary, everyday nature of its characters. Chopin's characters are not Dickensian caricatures, great men or women, or melodramatic stock types. Instead, they are nuanced, even restrained, individuals the reader could see existing in the real world.
When the reader is introduced to Mr. Pontellier in the first chapter, he is not presented as a mustache-twirling villain who mistreats his wife. His description as a neat, unexceptional man of forty and his mild manner as he smokes and reads the paper show him as a normal man of the period. The conflict between himself and the protagonist Edna is introduced in a subtle way, with his mild disapproval of her cavorting on the beach and burning her skin. Chopin says he observes "his wife as one looks at a valuable piece of personal property which has suffered some damage." This is a quick, realistic way of introducing not only marital tensions, but also the way men in this society view women at large.
Later on, when Mr. Pontellier reprimands Edna for not looking after their children enough, his behavior is portrayed not as evil, but as being very much a product of nineteenth-century ideas about gender roles. It does not occur to him to take care of the children, because that is what the wife does. As a husband, he believes it is his job to concern himself with making a living at his brokerage business.
Edna is also realistically portrayed. She is not a larger-than-life rebel or victim but a normal person living a life of quiet desperation. The discontent she feels in her life is not the result of over-the-top tyrannical male domination, but the result of society making her believe she must be a wife and mother, even though she finds no personal fulfillment in these roles. Little details, such as Edna's children not coming to her when they hurt themselves at play, reveal that Edna is not a natural when it comes to motherhood. Such details are realistic and allow Chopin's characters to feel authentic both socially and emotionally.
Kate Chopin is a writer of a genre of fiction called regionalism, or local color. This kind of literature falls within the realm of literary realism because it focuses on accurately representing the people, customs, and speech of a particular geographic region. Chopin, herself, was not Creole, but she married a Creole, and she very often writes about this group of people. In the first lines, the narrator describes Madame Lebrun's pet parrot who speaks a mixture of French, Spanish, and English, like the Creoles themselves. There are also realistic descriptions of the characters:
Mr. Pontellier wore eye-glasses. He was a man of forty, of medium height and rather slender build; he stooped a little. His hair was brown and straight, parted on one side. His beard was neatly and closely trimmed.
He is neat and trim, like most of the Creole men in this community. He is rather serious, however, and he looks "at his wife as one looks at a valuable piece of personal property which has suffered some damage." People are, thus, described quite realistically, both positively and negatively.
The Awakening is a novel of manners; by that I mean it's a novel more about how things are done and what people are feeling and thinking and the social construct (rules and behaviors) of the time. That being said, there are lots of realistic elements in the entire novel but particularly the beginning, for it sets the external stage for all the internal conflict which is to come. The most realistic element in those chapters is the sensory imagery--the sights and sounds and smells of the people and places.
The setting is a beach community, so we have the sights and sounds and feel and taste of beach--sand and sun and water. In the heat of the day, people are fanning themselves, trying to stay cool. We have "a number of bath-houses along the beach, of rough but solid construction" as well as cottages.
The cottages were all dark. A single faint light gleamed out from the hallway of the house. There was no sound abroad except the hooting of an old owl in the top of a water-oak, and the everlasting voice of the sea, that was not uplifted at that soft hour. It broke like a mournful lullaby upon the night.
Then we have clothing which sometimes rustles and the pipe smoke which has its own scent.
She wore a cool muslin that morning - white, with a waving vertical line of brown running through it; also a white linen collar and the big straw hat which she had taken from the peg outside the door. The hat rested any way on her yellow-brown hair, that waved a little, was heavy, and clung close to her head.
We feel somehow as if we're there when we can experience the story through our senses. It's these sensory images which allow readers to feel as if they're participating in the story.