It refers to the strange phenomonon of the side of the pillow that's facing the bed being cooler and more comfortable than the side that your head is on, no matter how many times you turn the pillow over.

This effect is probably due to the fact that the "other side" has no warm head to keep it hot, and it's just sitting on a mattress that's full of cool air.

This saying was, as far as I know, coined on ESPN, however it's since spread to other media, such as my daily speech.