Thinking they are in the Safari Zone, Ash tries to catch a Tauros, but is stopped by a Growlithe. Growlithe belongs to Lara Laramie, a Ponyta riding girl whose family owns the ranch - which is NOT the Safari Zone, they just raise lots of Pokémon. There is to be a big Pokémon race tomorrow to decide who gets 'in with the Laramie clan', whatever implications that holds. Lara will enter on her Ponyta, but an accident causes her to fall and break her arm.
Lara asks Ash to enter the race for her, riding Ponyta, because she really wants it to win and prove itself. Also, Lara's mean rival, Dario, will be racing with his Dodrio and she wants to beat him. So Ash enters the race with Ponyta, Misty with Starmie, Pikachu with Squirtle and Brock with Onix. They ride their Pokémon strongly, although all around them, their competitors and other racers are falling... why? Team Rocket, hiding along the track, create pitfalls and other difficulties to stall the other trainers, so that Dario, who hired them, can win. It was also Meowth who was responsible for Lara's accident in breaking her arm.
Despite everything, Ash manages to catch up to Dario and his Dodrio. Ponyta, through evolving into Rapidash, just manages to surge ahead and Ash wins, by a hair, after a tense buildup. (But did you think the writers would REALLY let Dario win? ^_^.)