Old hipster
slang for giving yourself some time to let the
alcohol out of your system; you'd go "on the wagon", either temporarily (if you were a heavy drinker or binge drinker) or one-day-at-a-time permanently ("Hi, my name's
Etaoin Shrdlu, and I'm an
alcoholic...").
It can also be used in reference to any drug, since our options have expanded in the subsequent decades. As has our vocabulary: rehab is a supervised form of drying out, and there's now an industry devoted to that supervision.