I did not see the segment, so I'm missing a lot oc clues. But some cyclists warm up and as part of the process take oxygen consumption measurments. Not positive why, perhaps to measure effort levels and do the right amount of warmup. In any case if oxygen consumption measurement is part of the process then the cotton is used to make sure that all the air taken in is through the mouth and when the measurements are being taken that means measured because if comes through a tube into a mask that covers the mouth.
Again I did not see the segment so I could be 100% wrong, but this is a possibility.