Car Masteries are perks you can apply to each car you own. Each one improves your gameplay experience in some way while you drive that car, such as earning extra Influence while you race, giving you a free Wheelspin etc.
Each car has its own skill tree, with some cars having more Car Masteries to unlock than others. Each car has a rarity attached to it; the cheaper, more common cars will have fewer Car Masteries (usually 8-10) to unlock than the rare, Legendary cars (up to 20).
Car Masteries are unlocked by spending Skill Points on a car; this is done through the [Cars] menu when you pause the game.
Depending on the car you will need 40-50 Skill Points in order to afford all of the Car Masteries available.