From the Super Mario Wiki
Before the event starts, Kamek flies over the course. He pulls out a telescope and spots Toad at the goal line wanting some Yoshi eggs. Then, Kamek spots the players on their horses at the starting line, and he then intervenes.
Like the regular equestrian event, players, as a team, can whip their horse for a boost in speed and jump. Players are trying to get to the end with the fastest time while also avoiding dropping the Yoshi eggs and dodging obstacles. The eggs contains White, Black, Pink, Yellow, Light Blue, Blue, and Red Yoshis.
In this event, teams race wagons to avoid dropping eggs and compete to reach the goal in the fastest time.