You wickets would be stickey too if you drank all that whiskey!
This is what wiki says:
The term sticky wicket comes from a situation the pitch has become damp, typically due to rain or high humidity. This makes the path of the ball more unpredictable thus making the job of defending the stumps that much more difficult. The full phrase is thought to have originally been “to bat on a sticky wicket.” Such pitches were commonplace at all levels of the game (i.e. up to Test Match level) until the late 1950s.
As a cricketer, I can tell you what I have heard.
wickets are the three sticks with two little sticks on top of them that are behind the batsman. Think of a small target between the batter and catcher in baseball. If the bowler (think pitcher) hits that target, the batter is out. Thus, the batter’s main job is to protect this target from being hit. If there is a possibility of running, all the better, but not needed.
When the ball barely touches the wicket, the bails (the two little sticks on top of the three bigger sticks-stumps) fly off, and the batman is out.
What I was told is that both bails and stumps used to be varnished to protect against the elements. When the sport spread to the colonies, India to be specific, the varnish would get tacky in the heat not usually found in jolly old home. The wicket being touched by the ball did not have the effect of bails flying (and having seen that sight many times from the crouched position of wicket keeper-it is really a great feeling) many sportsmen were put out (or not as it were) by the sticky wicket.
Choose whichever version you like, or make up your own.