Based on the information that the narrator gives readers, I think a reader could say that Smiley is mighty proud of his frog for two reasons. The first reason Smiley is proud of his frog is because his frog is an amazing jumper. Readers are told that the frog is "so gifted" and that the frog's strong suit is jumping on level ground. He could cover a great distance in a very short amount of time, and other animals of his breed are slower in those conditions. Smiley is so confident in his frog in those conditions that he always raises his bets. The second reason that Smiley is proud is because his frog has become famous. Men from "everywheres" have heard about the frog. They spread the news that his frog "laid over any frog that ever they see."

Here's what Twain says about Smiley and his frog:

You never see a frog so modest and straightfor'ard as he was, for all he was so gifted. And when it come to fair and square jumping on a dead level, he could get over more ground at one straddle than any animal of his breed you ever see. Jumping on a dead level was his strong suit, you understand; and when it come to that, Smiley would ante up money on him as long as he had a red. Smiley was monstrous proud of his frog, and well he might be, for fellers that had traveled and been everywheres all said he laid over any frog that ever they see.

Smiley trained his frog to jump so well that no other frog could outjump it. Being fond of making bets, Smiley was very pleased to have a sure winner. His pride probably grew as his pockets filled with money. Until the fateful day the stranger came to town!