Corribus' answer is just too clear, I have to add my mess...
A particle like 19th century physicists imagined them would fall on the proton. The electron doesn't because it's a wave, and waves occupy some volume. This problem of the atom's size was the reason to claim "electrons are waves" and introduce the corresponding theory of quantum mechanics.
The kinetic energy of a particle is bigger when the wave changes over a shorter distance. This is natural for an electron moving at uniform speed, it is less if applying the formula when the electron is "immobile", when the wave's amplitude doesn't depend on the time. Quantum mechanics says "stationary" instead, a more differentiated notion, where the electron can occupy all the time the same volume, hence radiate no light, but have a kinetic energy, an orbital momentum, an orbital magnetic momentum.
The kinetic energy maintains an optimum volume for the electron in the atom. If the wave occupies less volume, it changes over a smaller distance, and the kinetic energy increases. Below a certain volume, the kinetic energy increases more quickly than the energy of the electrostatic attraction decreases: this volume gives the smallest energy to the atom, and it is the volume that the electron occupies, it is the atom's volume.
"Volume" would deserve a definition, because the wave extends without limit, it only decreases to a negligible amplitude, so wording like "the amplitude falls below this arbitrary limit outside the volume" would be better.