Realism was the tendency of writers and artists in Mark Twain's era (the late 1800s to early 1900s) to create work that reflected real life, without romanticizing anything or making events too fanciful. To that end, this story certainly represents a semi-realistic view of life.
For one, it is centered in a very small county in Missouri about which there is absolutely nothing extraordinary, lavish, or extravagant. One aspect of realism is that it doesn't shy away from poverty or the lower classes, and in this story, Twain likewise focuses on a working-class, rather bored county.
Additionally, the dialogue reflects the common vernacular of the day, instead of utilizing "proper" English language. This dialogue reinforces the realism inherent in the story, by showing the characters as real people with realistic speech.
Finally, the characters are more important to the story than the events themselves, which is common in realist literature. The events are entertaining, but they are not the necessary aspects of the story. Realism focuses, instead, on the individuals and chooses to show how they act and react in these situations to present a realistic view of humanity.
Four characteristics of Realism that are evident in Mark Twain's story are that the action is plausible, the characters are more important than plot, the characters are portrayed with their real motives and temperaments, and the dialogue reflects vernacular speech.
The action of the story is completely plausible. A man from the East, when visiting a Western mining camp, asks about someone his friend told him to inquire about. He gets buttonholed by a talkative storekeeper and has to worm his way out of the situation.
The characters in the story are more important than plot. The story does not have much of an arc to speak of, except as explained in the above paragraph. The story inside the frame story is a series of vignettes about Jim Smiley's gambling. What makes the story interesting are the characters of Simon Wheeler and Jim Smiley.
The motives and actions of both Wheeler and Smiley are presented realistically with regard to their temperaments. Wheeler is addicted to talking and storytelling, so his capture of the narrator is understandable. The narrator's friend expected Wheeler to act just as he did, which resulted in the narrator's friend's joke working perfectly. Smiley is addicted to gambling, so his actions are understandable given his penchant for betting.
The most notable characteristic of "The Celebrated Jumping Frog of Calaveras County" that identifies it as a work of Realism is its vernacular diction. Wheeler's colorful way of telling the story accurately captures the speech of a man of that social class in that regional setting in that time period of American history.
Because of its plausible story arc that is secondary to character development, its characters who act consistently with their temperaments, and its "local color" in the form of Simon Wheeler's speech, Mark Twain's story is an excellent example of American literary Realism.
American Realism of the late nineteenth century was often satirical, and Twain's subject in this story is the assumptions that people make about one another based on where they live or were born or their social class. Twain often sought, through his work, to debunk stereotypes and attack bigotry. Thus Simon Wheeler, a rustic Western frontier type some might assume to be simple and slow, is able to outwit the sophisticated Easterner who narrates the story. American Realists often used vernacular speech in their true-to-life portrayals of ordinary people going about the business of everyday life, and that is certainly true of Simon Wheeler, who begins to relate the tale of Jim Smiley:
"There was a feller here once... in the winter of '49 or may be it was the spring of '50 I don't recollect exactly, somehow, though what makes me think it was one or the other is because I remember the big flume wasn't finished when he first came to the camp; but any way, he was the curiosest man about always betting on any thing that turned up you ever see, if he could get any body to bet on the other side; and if he couldn't, he'd change sides."
While Twain does use hyperbole to build humor, the outer frame of the story is plausible: an outsider visits a mining camp and encounters a raconteur who offers some local color and a practical joke.