The phrase sounds like nonsense until you trace it back far enough. In medieval Europe, a dog bite was a genuine terror, since it could mean rabies, and rabies meant death. 

One folk remedy held that the cure lay in the same animal that caused the wound: pluck a few hairs from the offending dog and press them into the bite. The logic was a kind of sympathetic magic, the belief that whatever caused an injury could also reverse it.

The remedy did nothing, medically speaking. But the phrase outlived the practice. By the 1500s, English drinkers had borrowed it for a very different kind of bite: the hangover. The cure, naturally, was another drink from the same "dog," that is, more alcohol.

Science backs the original idea about as well as it backs the modern one. A morning drink doesn't cure a hangover; it just delays the withdrawal your body is already working through. 

The hair of the dog was never medicine. It was optimism, dressed up as folklore, in both centuries.