Before explaining how the heiligenschein works, I will discuss some properties of animals's eyes that are closely connected to the heiligenschein. The photo below illustrates how a cat's eyes glow when light is shone on them.
Light rays from a distant source can enter an eye and be brought to focus on the retina as shown schematically below.
If that illuminated spot on the retina acts as a secondary source of light, the light that is scattered from that spot and returns back through the front of the eye will exit in the same direction it entered, headed back toward the distant light source. In many nocturnal vertebrates, the retina has a special reflective layer called the tapetum lucidum that acts almost like a mirror at the backs of their eyes. This surface reflects light outward and thereby allows a second chance for its absorption by visual pigments at very low light intensities. Tapeta lucida produce the familiar eye shine of nocturnal animals.
Humans don't have this tapetum lucidum layer in their retinas. If you shine a flashlight in a person's eyes at night, you don't see any sort of reflection. The flash on a camera is bright enough, however, to cause a reflection off of the retina and what you see is the red colour from the blood vessels nourishing the eye.
Many cameras have a "red eye reduction" feature. In these cameras the flash goes off twice - once right before the picture is taken, and then again to actually take the picture. The first flash causes people's pupils to contract, significantly reducing red eye. Another trick is to turn on all the lights in the room which also contracts the pupil.
Another way to reduce or eliminate red eye in pictures is to move the flash away from the lens. On most small cameras the flash is only an inch or two away from the lens, so the reflection comes right back into the lens and shows up on film. If you can detach the flash and hold it several feet away from the lens, this will help reduce the chance of redeye. Or try bouncing the flash off the ceiling if that is an option.