When talking about chemical solutions, the term supersaturated means that a solution contains more of a solute than can be be dissolved in it. This usually makes some of the solute separate itself from the solution.
Rain happens in a supersaturated solution of water in air. Carbonated water is a supersaturated solution of carbon dioxide gas in water. At the start, the pressure in the bottle is very high. This means that more Carbon dioxide gas can be put into the water than at normal pressure. When the bottle is opened, the pressure falls to the normal level, and the carbon dioxide gas escapes very slowly. Certain things such as small bubbles in the bottle can make this process faster. Such bubbles can be caused by shaking the bottle. Another solute such as sugar powder or a widget can also trigger precipitation.