Why does kinetic energy quadruple if speed doubles?

Work (which is a type of energy transfer) = force x distance.Look at the formulae for acceleration. The relevant formula in this case (simplified for the case that the initial speed is zero) is:

Distance = (1/2) acceleration time squared

Let's keep the acceleration constant. Note that this will also make the force constant. In this case, in twice the time your object will move four times the distance. Note that at constant acceleration, in twice the time you get twice the speed (assuming that you start at speed = 0).

Combining all of the above, to get twice the speed, you need to apply the same acceleration (and therefore, the same force) over FOUR times the distance. As a result, the amount of work required to achieve this acceleration will also increase by a factor four. Or in general, it will scale with the square of the speed.