Mikes' answer is the most intuitive, but he rephrases the question in a possibly non intuitive way.
Odd that nobody mentioned power, which scales linearly with speed. Of course if you add linear increasing amounts of power to the system the energy will increase quadratically.
Power scaling linearly is more intuitive because doubling your speed requires twice the power to maintain the same force, why does it require twice the power? because you have half the time to power it.
This is basically it. If power is linear with respect to velocity, then it becomes illogical for kinetic energy to be linear with respect to velocity.
The energy of the object is simply the integral of power over time and that happens to be a quadratic function.