It's caused by the fact that the stars themselves are much smaller than the resolving power of the telescope; for this reason, they can't be focused to a single point. It's called the "diffraction limit."
If you look closely, you can see that many of the points of light in the image do not have the "flares," these are entire galaxies, and so have (angular) size closer to what Webb can resolve.
The flares themselves are actually images - in "inverse space," the spatial frequency domain - of some part of the optical train between the main mirror and the image sensing array. Possibly the entrance aperture, which is rectangular, or some filter holder.
You made my head explode. Thanks for that.