Why the electron occupies only discrete levels is understood by considering the electron to be, not a particle, but a wave.

de Broglie's Matter Wave * A wave is associated with every particle and that the wavelength of a matter wave is inversely related to a particle's momentum. * Matter waves can be reflected, refracted, diffracted, and cause to interfere. * The discrete values of radii of Bohr's orbits are a natural consequence of standing electron waves. The electron orbits in an atom have discrete radii because the circumferences of the orbits are integral multiples of the electron wave lengths. The wavelengths differ for different elements (and also for different orbits within the elements), resulting in discrete orbital radii, or energy levels, which are characteristic of each element. 