From what I understand, it's a simple procedure done to test for presence of tumors, cataracts and other deformities within the eye. A light shined into a healthy eye should reflect off of the retina in the back of the eye and give off a red reflection (thus the "red" in "red eye reflex"--also the same effect seen in red eye photographs). If there is anything abnormal within the vitreous of the eye, the light will reflect off of that instead, giving off a white reflection.