How does Jim bet on his mare in "The Celebrated Jumping Frog of Calaveras County"?
Jim enters his mare into races, fooling others into thinking she can't possibly win, and he places bets on her.
Because the mare looks very slow and sickly, people don't believe she'll win. (When others are very confident that your horse will lose, then you actually get more money when your horse wins than if your fellow betters had more confidence in your horse.) So, his mare looks very ill and starts off going very slowly in the races. Her unfortunate appearance and gait have earned her the nickname "the fifteen-minute nag," meaning it takes her fifteen minutes to go once around the track. (In other words, since the track is 1/4 of a mile long, she only goes one mile an hour.)
So, Jim's mare is such a poor competitor that she even gets to have a head start in the races. At first, as the race begins, she does horribly. The other horses pass her by as if she never had the head start.
But then suddenly, toward the very end of the race, she puts on a burst of speed and makes this awkward, messy, loud bolt for the finish line--and then she wins, just barely beating the other horses.
The man who retells this story, Simon Wheeler, says that Jim is just lucky and that's why he wins his bets a lot. But as readers, we know a bit better. Jim is tricky. He lets people think that his mare is not a good competitor; he places the bets, and then he collects his winnings. He did the same thing, basically, with his dog (entering him in fights) and his frog (entering him in jumping contests).