There are two reasons Twain goes through Smiley's other modes of betting before getting to the famous frog of Calaveras County. The first is to provide the substance for the satire of the story. Besides being an amusing short story--unless one is of the same opinion as the button-holed inquirer after Reverend Leonidas W. Smiley--it is a satire upon the habit of betting in the West, or at least in Calaveras County. Satire is a jesting, ridiculing look at an element of a culture or an individual or a group of individuals for the purpose of exposing just how wrong-headed the behavior or individual or group is. In this case, Twain is satirizing those who are uneducated and who freely, incessantly bet.
Therefore, part of Twain's ploy in establishing an effective satire--the purpose of which is to inspire a change in the behavior or situation or individual(s)--is to make this type of betting appear to be as exaggerated and absurd as he possibly can. Consequently, he goes on and on and on about how Smiley would bet, "he'd change sides," and on what he would bet, "rat-tarriers, and chicken cocks, and tom- cats, ... till you couldn't rest," and about incidents of betting,
Parson Walker's wife laid very sick once, .... [He said] with the blessing of Providence, she'd get well yet; and Smiley, before he thought, says, "Well, I'll risk two- and-a-half that she don't, any way."
So the mare and the pup receive detailed telling yet are not the first to be highlighted. The list of which they are a part (as I can sort it out) includes betting on:
- two birds setting on a fence,
- Parson Walker at camp-meeting,
- straddle-bug (hypothetically),
- Parson Walker's sick wife,
- the fifteen-minute nag,
- a little small bull pup,
- and the frog
The other reason Twain gives details about the mare and the pup before getting to the frog is to establish characterization. Now, this is an odd point. Characterization isn't needed because Smiley undergoes any changes or epiphanies for which careful characterization is needed. Twain devotes all this attention to Smiley's characterization for two reasons: (1) to make the tale as "monotonous ... interminable ... absurd ..." as possible;
the monotonous narrative ... he never changed his voice from the gentle-flowing key to which he tuned the initial sentence [of] the interminable narrative ... such a queer yarn [told] without ever smiling, was exquisitely absurd ....
(2) to make his satirical points as emphatic as possible. In other words, had Twain jumped right in with Smiley getting straight to the frog incident, there would have been less effectiveness to the satire and there would have been less exasperated amusement at the whole amazing tale Smiley tells. By giving details on the mare and pup first, Twain fulfills his objects for writing the satirical story about betting on the frog of Calaveras County.
He want to build up and establish Smiey's obsession and habitual behaviour of dwelling into gambling with money in the casino and after that, build up and carry on the main story again
He wants to show that you can't always judge something by how it looks, and that you never know how it will turn out. He's also trying to show how Smiley was always able to come up with a great trick when it comes to animals and betting. He's setting up the story with some less fantastic examples while building up to the frog.