BHS -> Staff -> Mr. Stanbrough -> Physics -> Mechanics -> Kinematics -> this page
Without any calculation at all, you can see that if the speed of the object increases with time as it falls, the distance that it falls each second must increase as it falls - after all, it is going faster and faster, so it will cover more and more distance in the same time interval.
To see how the distance fallen in free fall increases with time, study the animation at the right.