Readers should look to the fourth paragraph in order to find evidence to answer this question. The paragraph is somewhat lengthy with examples, but the main gist of why Smiley is the "curiosest man" is because Smiley is a compulsive gambler. He's got an addiction. The narrator doesn't mean that Smiley is a guy that wonders about stuff. In this case, curious is being used as a synonym for "odd." Smiley is odd because he bets on anything and everything. What's also strange is that Smiley will take the other side of a bet just to have a bet with/against someone. Then, strangest of all, Smiley somehow manages to win far more bets than he loses. Smiley bets on which bird will leave a fence first, which dog will win in a fight, and how long it will take a bug to get to its destination.
If there was a...
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horse-race, you'd find him flush, or you'd find him busted at the end of it; if there was a dog-fight, he'd bet on it; if there was a cat-fight, he'd bet on it; if there was a chicken-fight, he'd bet on it; why, if there was two birds setting on a fence, he would bet you which one would fly first; or if there was a camp-meeting, he would be there reg'lar, to bet on Parson Walker, which he judged to be the best exhorter about here, and so he was, too, and a good man. If he even seen a straddle-bug start to go anywheres, he would bet you how long it would take him to get wherever he was going to, and if you took him up, he would foller that straddle-bug to Mexico but what he would find out where he was bound for and how long he was on the road.
The exact quote from the story is:
he was the curiousest man about always betting on anything that turned up you ever see, if he could get anybody to bet on the other side; and if he couldn't he'd change sides. Any way that suited the other man would suit him any way just so's he got a bet, he was satisfied. But still he was lucky, uncommon lucky; he most always come out winner.
Why does Wheeler call Smiley the "curiosest" man?
Simon Wheeler, the garrulous narrator of the story-within-a-story in Mark Twain's "The Celebrated Jumping Frog of Calaveras County," describes Jim Smiley as:
the curiosest man about always betting on anything that turned up you ever see, if he could get anybody to bet on the other side.
With this, he launches his long-winded account of the man whom the narrator of the frame story doesn't even want to know about.
The trait that made Jim Smiley such an anomaly—"the curiosest man"—was his addiction to gambling. Wheeler recounts Smiley placing bets on how long it would take a "straddle-bug" to reach its destination, which bird would fly from a perch first, and whether the minister's wife would recover from her illness. Smiley didn't much care which side he took in such contests, but he won more often than not.
Wheeler then goes on to describe in more detail Smiley's "fifteen-minute nag," which, despite its asthma, always finished a race a neck ahead of the next horse. Then Wheeler waxes eloquent (in his miner's dialect) about Smiley's bull-pup named Andrew Jackson—which didn't look like much of a fighter but could overpower any dog that had hind legs. Finally, Wheeler gets to the main story of his tale, which is about the frog named Dan'l Webster.
Wheeler could have (and would have) gone on and on about the unusual man, if the narrator hadn't made his escape at the first opportunity. But all his stories served to reinforce one idea about Jim Smiley: "He'd bet on any thing—the dangdest feller."
The major reason that Wheeler gives for Smiley being so "curious" is the penchant he has for betting on any thing that comes along. But it is coupled with the fact that, unlike most people with such an addiction, Smiley seems to be able to find a way to win many of the bets he makes. And many of these wins come in the most outrageous fashion.
Wheeler goes on to describe the menagerie that Smiley had, including his fifteen minute nag who would find a way to spastically work herself back into a race and beat horses she had no business meeting. The menagerie also included a bulldog with no hind legs named Andrew Jackson who won many a dog-fight against abler, stronger and meaner dogs. It was simply the most stubborn dog around once it got its jaws locked onto another dog.
Wheeler also recounts that Smiley caught a frog and then took him home to "edercate" him and help him learn to be a champion jumper. All of these would likely make him "curious" in the eyes of Wheeler or anyone else for that matter.