To me, the humor in this story is mostly in the absurdity of the things that Simon Wheeler is saying. The humor is not subtle -- it's just really goofy.
For example, I really like the story of the dog that would win its fights by grabbing the other dog's hind leg and never letting go. The idea that the dog would die of a broken heart when it fought a dog with no hind legs is just ridiculous. To me, it's the absurdity of the situation that is funny there.
In other places, what's funny is the wording of the story. I think, for example, that this description of the frog is pretty funny
You never see a frog so modest and straightforward as he was, for all he was so gifted.
The idea that a frog could be modest is pretty funny because that just isn't a characteristic a frog could have.
I hope that helps... it's not easy to explain why something is funny.
The narrator of the inner stories in this Mark Twain classic, Simon Wheeler, tells a string of anecdotes that run into each other—all related to the gambling habit of Jim Smiley. All are humorous, even the idea of betting on which preacher at a camp meeting would be the "best exhorter." The story about the horse and the one about the dog each relies on dialect, imagery, and irony for its humorous effect.
The story about Smiley's "fifteen-minute nag" uses humorous dialect in the following phrases: "for all she was so slow and had the asthma," "she'd get excited and desperate-like," and "always fetch up at the stand just about a neck ahead, as near as you could cipher it down." These quirks of language reveal the narrator's lack of education and folksy style, which add interest and color to his stories. The sight and sound imagery of the nag kicking, coughing, sneezing, and blowing its nose brings a very funny picture to mind, like a cartoon. The fact that this apparently asthmatic and ungainly horse can consistently win its races is surprising, which adds to the humor.
Smiley's bull pup named Andrew Jackson was likewise an unlikely winner, but the irony hinges on how the pup loses rather than on how it wins. Andrew Jackson is unable to win against a dog with no hind legs because Andrew Jackson's "pet holt," or favorite place to bite, was the hind legs. Wheeler tells the story descriptively so we can picture the bull pup reaching for its favorite spot to bite, finding no legs, looking betrayed, and slinking off to die. Again Wheeler's dialect adds to the humor with phrases like "he got shucked out bad," throwed in the sponge," "looked sorter discouraged-like," and "would have made a name for hisself if he'd lived." There is no denying that this anecdote also relies on a type of humor called dark humor, gallows humor, or grotesque humor. In such humor, a victim is mocked or something that one should not laugh at is made fun of. As much as readers want to feel sorry for the opposing two-legged dog whose legs were "sawed off in a circular saw" and for Andrew Jackson, who "limped off a piece and laid down and died," they find themselves snickering and possibly even guffawing at the story because of the way Wheeler tells it.
The stories related by Simon Wheeler are hilarious because they use imagery that creates outrageous visual pictures, dialect that is unusual and colorful, and surprising situations. Twain is even able to pull off gallows humor in the form of mocking a disabled dog and the death of another dog through the side-splitting narration of old Simon Wheeler.