This common, but unusual saying is popularly made in reference to curing a hangover, and comes from the idea that "like cures like"- in other words, drinking more of what made you ill in the first place will actually make you better! But what do dogs have to do with hangovers?
The phrase actually stems from the belief that if you were bitten by a rabid dog, placing a hair of the dog that bit you into the wound would prevent you from getting rabies.
Needless to say, neither of these remedies has any medical basis, however a Bloody Mary the morning after a long night out continues to be a popular treatment for a hangover!