This trophy requires you to start a race and simply stay driving close behind one other car for the entire race. Drafting is where you are directly behind another car and the car in front of you is pushing all of the wind out of your way so that you can (usually) get a good boost in speed and pass them. While obtaining this trophy, you don't want to pass the car in front of you, just stay close behind them and follow their movements. Brake when they brake and accelerate when they do. 

It's recommended that you choose a long race (usually with 3 or 4 laps) with slower cars so that the car you're following behind doesn't just take off ahead of you where you can't keep up half the time. A recommended race to attempt this trophy on is Mountain Dash, the second race in the tour. This event has 3 laps along with a lot of turns and a small number of straight-aways, which means that the car in front of you shouldn't get too far away from you at any given time.

One last thing to note is that these are cumulative points in one event. You don't have to stay directly behind another vehicle for 2,187 straight yards (2000 meters), you just have to have that much drafting distance over the entire course of that event.