Why does a siren change pitch as it passes?

Because the waves are bunched up ahead of the source and stretched out behind it — the Doppler effect, which is one of the few pieces of physics you can hear directly.

The mechanism. A siren emits sound waves at a fixed rate. As the vehicle moves towards you, each successive wave is emitted from slightly closer, so the waves arrive more frequently than they were emitted — a higher pitch. Once it passes, each wave is emitted from slightly further away, so they arrive less frequently — a lower pitch. The siren itself never changes; the change is entirely in the geometry between it and your ear.

Why the drop happens so abruptly. The shift depends on the speed directly towards or away from you. When the vehicle is approaching head-on that component is the full speed; as it passes it swings through zero and reverses within a second or two. The pitch therefore slides down rapidly at the moment of passing rather than changing gradually, which is exactly what you hear.

Why the driver hears no change at all — they are not moving relative to the siren.

Where else it appears:

Astronomy. Light from receding objects is shifted towards red, and this is how stellar motion, galaxy rotation and the expansion of the universe are measured. Most known exoplanets were found by detecting the tiny Doppler wobble a planet induces in its star.

Weather radar, which measures the shift in returned pulses to see how fast rain is moving towards or away — how rotation inside a storm is detected.

Medical ultrasound, measuring blood flow by the shift off moving cells.

Speed cameras and radar guns.

Where it does not apply. A stationary sound whose pitch changes is doing something else — that is the siren's own pattern, wind, or the way reflections arrive.

And if the source exceeds the speed of sound, the waves pile into a single shock front: a sonic boom, which is a continuous phenomenon, not a one-off bang at the moment of breaking through.