A curveball "breaks," or curves, in the air in order to rapidly change the trajectory of the ball, thus in theory making it harder for the batter to hit. Kershaw's 12/6 curveball can break up to several feet. Kershaw might have had a tough season but nothing changes the sheer absurdity of his curveball.
So, what makes a ball curve in air then?
Curve is actually an interesting and peculiar consequence of the Bernoulli effect. Consider a baseball that just left the pitcher's hand, and is traveling in air toward the catcher. If we ignore forces due to gravity and air resistance, some tertiary force would need to exist in order to alter the direction of the baseball's motion. This force, dubbed the Magnus force, is the result of pressure differentials on the two different sides of a spinning baseball.
As a baseball spins, it carries a very thin "boundary layer" of air along with it. This boundary layer of air is thus spinning with the same rotational velocity as the baseball. As the baseball continues along its path of motion, the spinning of the boundary layer of air accelerates the air on one side and decelerates the air on the other side, creating a pressure differential. We know from the Bernoulli effect that fluids traveling at high velocities must exist at lower pressure states, and fluids traveling at comparatively lower velocities must exist at higher pressure states.
The differential in pressure causes a net force imbalance on both sides of the baseball. The side of the ball that experienced a net deceleration of the air around it has a higher air pressure, and thus a greater net force exerted upon the ball by the air. This is the Magnus force, that "tertiary" force which causes the ball to curve in the air (Flift in the below image).
Thus, the more extreme of a spin imparted onto a baseball, the more extreme of a curve it will assume in air.
With the Bernoulli effect in mind, it's pretty easy to appreciate how Kershaw generates such a ridiculous curveball.