I suppose my interpretation of his message is that if you think of a “ray of light” that is not a mono frequency laser as an “input signal”, a convex lens will smear it out in to its constituent component frequencies. From that perspective you can analyze the original signal (aka color) with a geometrically/spatially separated spectrum of values.

It was a single spatial point of ray intersection with your sensor or eye. You’d need color filters/retinal cells to pick apart the frequencies in the complex waveform.

After rainbow separation, the components are spread across multiple sensors giving you a frequency domain view.

Aside from cranking the math, here's how I think about it: in the far field of a small aperture, the electric field has spherical phase (think expanding circles), and the field distribution is the Fourier transform of the aperture. A lens is an element that adds spherical phase - a plane wave passing through a convex lens now has a spherical phase distribution. So the lens focal point is now the tiny aperture in that system, and since the math works out the same no matter which way the light is going (reciprocity), the focal point is the FT of the field at the input of the lens.

Goodman is great, Hecht and Zajac covers more fun with optics at an intro level.

Yes the explanation is diffraction. As light passes through a lens, diffraction acts in similar way as a light through a small pinhole. Incidentally, pinholes and apertures are low pass filters.

Some more info here

Miles V. Klein, Thomas E. Furtak - Optics 2nd ed, Wiley

Joseph W. Goodman - Introduction to Fourier optics, W.H. Freeman

Upping a cousin comment.

https://www.youtube.com/watch?v=Y9FZ4igNxNA

Adding that, again, optics is a difficult and frustrating field. Don't worry that you're struggling to connect it. It took me a few years in an optics lab working hands on with light every day to really come around.