The Shapiro team animation is shown on the website IllusionContest.
However, when you look at it out of the corner of your eye (looking directly at the blue dot), then it appears to be moving at an angle. When a batter looks at the ball with his or her peripheral vision, it appears to move to the side.
The website states, “In baseball, a curveball creates a physical effect and a perceptual puzzle. The physical effect (the curve) arises because the ball’s rotation leads to a deflection in the ball’s path. The perceptual puzzle arises because the deflection is actually gradual but is often perceived as an abrupt change in direction (the break).”
And, “Our illusions suggest that the perceived “break” may be caused by the transition from the central visual system to the peripheral visual system. Like a curveball, the spinning disks in the illusions appear to abruptly change direction when an observer switches from foveal [central] to peripheral viewing.”
Thus, when the spinning ball is seen moving directly in front of one’s eyes (using our central visual system), it appears to move in a straight line.
However, eventually it is seen out of the corner of one’s eye (using our peripheral visual system), as if standing sideways in the batter’s box, and the spin of the ball makes the ball appear as if it is curving.
Page three talks more about the curveball study, and also provides the other finalists of the contest.