As you play through the game and earn Influence you will eventually be told of a Barn Find rumour, which adds a purple circle to your map.
Head to that point on the map and locate the barn (an abandoned shed which is shaped like half a cylinder on its side) to find the car.
After a certain amount of time (you may want to do a few races) you will be told that "your Barn Find is ready". Return to the Festival Hub or any homespace, navigate through the menus and collect your Barn Find once you locate it.
A tip for finding Barns:
While you're in the purple circle pause the game and select Drone Mode. Then, fly your drone within the designated area until you find the barn. Once you have, the circle will disappear and be replaced with the precise location of the barn, after which you can return to your car and drive to the location.
If you purchase the Treasure Map you will immediately be shown the barn location instead of the purple circle when a new rumour comes up.