If you watch the bottle being passed 48 seconds and 3:16 sec in to
this clipit will show what is meant.
A 'sticky bottle' is when the plastic water bottle (bidon) is passed from a car (usually their team car) to a rider and the rider holds on to it for a few (or quite a few!) seconds.
If you ever watch a race closely you will notice that a bidon is never just handed straight to a rider. The person in the car holds the bidon forward and the rider always grabs it by putting their hand back behind to grab it - both are now holding the bottle with force applied so the rider can avail of a small yet welcome respite.