This is where classical mechanics breaks down and quantum mechanics comes into play, and one of the basic concepts of quantum mechanics is that all matter has a wavelength dependent upon its mass and its velocity. You, me, a hammer, the planet Earth, and everything else in the Universe has a wavelength.
So why, then, don’t we notice? For the same reason we don’t notice atoms: Because we exist at a size well above the scale where the phenomenon becomes interesting. Our quantum effects cancel out, statistically speaking, and give us the illusion of matter and energy and a stable physical realm. It’s like how different wavelengths of sound cohere to form a tune.
At the level of electrons, however, the quantum effects come to the fore. The electrons have a wavelength, too, but a constrained one: It can only exist in a predefined number of states, defined by how energized the electron is. Each state has its own shell, so the electron’s state determines which shell it is in. It is impossible for the wavelength to exist at, say, 1.5 of a shell: It must pick 1 or 2, and it can never ever be a fraction as long as it is in orbit around a nucleus.
So, how is this accomplished? The quantum leap. When an electron changes states (by absorbing or releasing a photon), it performs a quantum leap to the next closest shell to reflect its leap to the next closest state. When an electron absorbs a photon, it jumps to the next higher state. When it emits a photon, it jumps down a rung.
Now, what is a photon? It is a quanta of electromagnetic energy. Like a photon, it has a wavelength determined by its energy level even though it is massless. (Mind-bending? Try the double-slit experiment. ;)) What’s more, it cannot split: You can never have a half of a photon, or an eighth, or two-thirds. Since electrons gain energy via photons (and lose energy the same way), electrons have no method of getting or losing anything less than a whole amount of energy.
In short, electrons are constrained as long as they are in orbit around a nucleus. The jumps are related to wave-particle duality, one of the most powerful concepts of quantum mechanics and one of the least intuitive. If you still can’t `see’ it, hang around and wait for some experts. But don’t expect it to be analogous to anything you can physically see around you: The best you can do is observe the effects.