Why does pulse-width modulation work?
Depends on what you are controlling. For DC-motors it is the inertia of the device that acts as a low-pass filter of the PWM modulated signal resulting in a continuous motion. For most LEDs it is the human eyes that do the apparent low-pass filtering.

If the PWM-frequency is not very high you can actually see this by moving your head from left to right while looking at the PWM modulated LED. The light will appear on and off while the LED is moving through your field of vision (this can get very annoying with LED car taillights).