The quartz crystal itself is an analog component which resonates at some specific frequency. The crystal is placed within a feedback circuit to create a stable, sinusoidal oscillation; the analog sinusoid is then converted into digital pulses to be counted.

It's the same principle as a "pendulum- or spring-driven clock mechanism, with discrete periodic movements"; just on a microscopic scale -- you're taking an analog physical system which naturally resonates at some specific frequency, and then converting the continuous motion of the system into discrete pulses.