IANAMathematician, but I've tried to come up with a metaphor which I hope isn't too wrong:

1. You've start with a signal fluctuating going up and down, and it's on a strip of little LEDs labeled from -1 to +1.

2. You mount that strip to a motor, and spin it at a certain rate. After a while the afterimages make a blob-shape.

3. For each rotation rate, measure how much the shape appears off-center.

In this way you can figure out how much the underlying signal does (or doesn't) harmonize with a given rotation hertz.

An interesting mechanical analogy! In this example, the "integration" is done by the eye and perhaps by the LEDs depending on their type. But it's more akin to the Gabor Transform than Fourier, as there is locality in time.

Sounds like a variation on the mechanical strobe tuner.