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 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.