A voltage is passed to the crystal and turned off, the crystal then UN-distorts back to it's original shape sending out a pulse, how does that affect the capacitor and the roles they play back to the Arduino Pins that have the xtal across them?
The crystal doesn't merely "un-distort", it "rings" like a bell, at a particular frequency. Like a plucked violin sting.
This gradually dies out without further stimulation, but if you were to operate at electronic speeds much faster than the oscillations themselves, you could detect the peaks of the ringing, and add additional energy at just the right times to keep the crystal vibrating at the same frequency (rather like the bow of a violin.)
At least, that's how I visualize it.