I don't know if this kind of visualizing Fourier transform is helpful to anyone doing/will do work using this idea, but it is not useful to me, and I don't think it is useful to anyone, apart from being somewhat interesting representation, in a way an illusion.

Here is how I see this:

-Sinus (and cosinus etc) is defined using a right triangle, unit length hypotenuse simplifies things, so they are shown on a right triangle inside a unit circle.

-Time dependent sinus function, sin(wt), can be represented using a uniform circular motion and y coordinate of position.

-Fourier says that you can represent (may be almost all) functions using sinus functions with proper frequencies and amplitudes. There is a way to find those coefficients using some mathematics: infinite or finite sums or integrals depending on application.

-Oh well, I have a wonderful idea, great analogy, listen: why not show Fourier transform/series using a circle on circle on circles, uniformly moving (constant speeds) particles and projections of their position to sideways as a function of time.

I don't think understanding its mathematical form is more complicated than understanding what a function is. It is already very simple, too much enforced simplification makes it complicated.