There's more CO2 dissolved in the water than there can be at atmospheric pressure. The CO2 is constantly trying to escape, but in order to do so it needs a nucleation site that disturbs the water. When the drink is shaken, lots of little bubbles form, and stick to the inner wall of the drink. These bubbles are nucleation sites. Flicking the side of the drink makes them float up and pop.

Increases pressure so the gases dissolve back into the liquid. Probably.