Why does a hexagonal aperture produce spikes in the image? I was reading this explanation as to why the JWST images contain six prominent spikes. It explains the effect with this image:

Considering only the single hexagon -- why does this aperture by itself produce six spikes?
(If I'm reading things correctly, I believe this question can be more technically phrased as "why is the point-spread function of a hexagon six spikes?" -- but I might be misunderstanding.)
 A: This pciture sequence is really a simplification of the real situation. The last image is not presented because there is a switching algorithm between the segments individual pictures and the the added up picture. The image sequence shows in the last picture the uncorrected result.
In the corrected result like seen in the official JWST pictures the first aperture is seen that comes with the spheric star itself. It is somewhat an self-aperture. That is on the other hand only not correctable for high intensities of the stars light.
That is the case with the second picture name Hexagon. It comes from all individual segments for strong, high intensity impact of star light this not compensated by an correction algorithm.
There is if you take a close look that the bright stars a bar

Credit.
that is additional to the sic spikes this stems from the three last pictures as you can see it fills from left to right and so it is a signature of the JWS telescope.
But there is a fine structure in the spikes as well. And there is a substructure close to the the circular apertures diffraction Airy pattern that is very spikey. That does not stem from one of the pictures in the sequence of Your question. This is because the diffraction limited pictures are not calculated over incident intensity.
This page shows how to transform the diffraction image into the function of light intensity with the order of diffraction with an laboratory example that show the spikeys already.
This show with simple examples what is behind this in mathematics with the methods of Mathematica: calculate the 2d fourier transform of an image. This is an example where they go a little bit deeper than needed into the problems: using 2d fourier transform of an image to detect typical wavelengths.
This shows up too that the significant dimensions stem from the apertures at works.
I assume that the published text of the source You referenced is reduced and simplified for a certain group of readers. The original work did include some of the remarks I pointed out and made clear. The spikeyness is for sure containing more intrinsics of the 1st mirror of JWST and the second aperture in the detector including the analog to digital converter and the properties of the used red light filters for contrast enhancements and line detection separation purposes.
The shown sequence is an introductory link and a set of clues how to understand and interpret the high intensity stars images. Mind that these are not with the other stars of low or moderate intensities.
The internal spikey ripples on the six spikes reflect too JWST geometry and parameters and detector sensivities. Seeing this is the surplus of the design and the manufacturing result and the intricates of the high price and long time spend on it. They show on the other hand that there is more intensity going into the strucuture of the spikeys than into the Airy patterns.
Yikes means something like yield of maturity. It is a compliment to the resolution of the picture in the intermediate range of intensity near the highest intensity. So it addresses the surfaces related resolution of the dynamic range of the analog radiation to digital signal system of the JWST. The peripheral structure of the corona of the imaged star is really another surplus seen in the first pictures. It can be read as opulent optics and imaging in astronomy.
