I've never heard this adage, though it appears to be "common knowledge." (Probably because I've never hosted Thanksgiving, and I will rarely roast a turkey with stuffing for my own personal dinner "rarely," of course, meaning "never.")
It turns out there are good, scientific reasons for not storing turkey and stuffing together. It's not exactly a dangerous situation there's no reaction that says "turkey + stuffing + refrigeration = apocalypse" but it increases the likelihood of food contamination.
As I'm sure you know, the reason food goes "bad" is because of the actions of bacteria and other microbes. These guys tend to be most active when the temperature is between 5 and 60°C (40 and 140°F) they thrive at room temperature, but they hate ovens and freezers, and they're not too fond of refrigerators either. That's why we cook food before eating it, and that's why we refrigerate leftovers. (Refrigerator interiors are around 1.5 to 3.5°C (35 to 38°F).)
One thing we have to remember is that while refrigeration does a good job of slowing down (but not stopping) microbial activity, it doesn't happen instantly. When you put something warm in your refrigerator, it doesn't immediately convert to 2°C. It takes time sometimes quite a while to cool down. And while the temperature of your cooling food is still in the "danger zone" for bacteria, the bacteria are going to continue acting and multiplying.
When you put your Thanksgiving turkey and stuffing in the refrigerator (which you should do within two hours of removal from the oven), you want them to cool down as quickly as possible. Stuffing, in particular, is a haven for bacteria, as it is moist and resistant to temperature change (slow to heat up, slow to cool down). If the turkey and stuffing are stored together particularly if the stuffing is still stuffed inside the turkey, which serves as a great insulator the cooling process can be slowed, leaving both items in the danger zone for an extended length of time. Separated, they cool down faster, giving bacteria less of an opportunity to operate.
See: Canadian Food Inspection Agency, CBC, Real Simple