The Backgammon Race unlocks at level 28. The Race grants you points for your performance in contests. Every time you rank in the top 500 of any contest throughout the duration of the race, you get race points.
The more times you rank in a contest, the more race points you get. Also, you get more points when you rank at the top of a contest.
To checkout, a Race, simply click on the contest tab on the right of the screen, then select the Race tab at the top right.