After you reach Level 2, you will start to see challenges from other drivers in the Beach Buggy Racing League.
When another driver challenges you, their picture will appear in one of the posters on the main Race screen. If you win the race with the driver, the driver will be unlocked and will join your team.
If you lose the driver challenge, you can retry it for gems, or you can wait for the driver to challenge you again for free. You can also look at the driver on the Team screen, and challenge the driver from there.
Upcoming driver challenges can be seen on the Profile popup. Click the button with your name on it in the upper left corner of the screen to see an enlarged version of your level progress, and look for the icons with the crowns.