I still don’t understand how that worked. At the time I thought it was “getting your eyes used to the bright light so they wouldn’t turn red with the big flash,” but that definitely doesn’t make sense.

I understood it as the red eyes you see in photos is the wide open iris of an eye you're photographing zooming in on the blood vessels in the back of the eye. Flashing bright light before the photo makes the iris of the person you're photographing contract significantly, so you can't see the blood vessels in the back of the eye anymore.