All you have to do is get yourself a car, find the car you want to race, and do a burnout near it. If the car is moving, don't worry, using the horn also works.
When you're sure the owner got the message, find a good place to start, and set up a waypoint where you want the finish line to be at (if you don't, a random point ahead will be chosen). Once the challenged car is on position, the race will start.
That's it!