“Corks in cycling vernacular date back at least to the 1920’s, probably originating at the steeply banked wood track indoor 6-Day Bike Races. During the halcyon years of cycling in the tens, twenties, and thirties, trackside at the 6-Day race was one of the places where the social set went to be seen. Sitting next to the action at their infield tables, it was chic for the “swells” to sip champagne and sine while giving the track stewards money for sprint laps" (Ernst, 2009).
Apparently the idea evolved into the idea that 'popping' a cork on a bottle released the energy inside in a burst of fizz and carbonation. Over some time, the champagne would start to go flat, eventually becoming entirely flat.
Thus, 'popping' a cork would come to mean something like 'a burst of energy' and cyclists took to hanging a cork on their saddle so that they would always have some energy left on the last sprint!