How is something carbonated?
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You know how you can press rubber duck underwater and as long as you hold it down, it will stay in the water, but when you let go it will pop out. Similarly the carbon under pressure can be pressed into water. But once the pressure is gone, the carbon molecules will start popping out. Rubber duck has to more do with physics, while carbonated things have to do with chemistry but the idea is the same.
In beer and cider, it’s a natural effect of the brewing process. The yeast is basically eating sugar, peeing out alcohol and farting out CO2.
In sodas, the carbonation is forced into the drink from a pressurized tank. Since the tank is at a higher pressure than the drink bottle, the CO2 has no choice but to go into the drink.
They pump carbon dioxide under pressure into the bottle.
As long as the bottle holds the pressure, the carbon dioxide will dissolve and react with water to form carbonic acid (CO2 + H2O -> H2CO3). This acid gives a slightly sour taste to the drink.
Anyway, if the pressure is removed, for example if you open the bottle and leave it open, the carbon dioxide will unreact with the water and fizz out, and eventually the drink will be “flat”.
If you shake the bottle, the process of “fizzing out” is sped up enormously.
If you shake the bottle while it’s still pressurized (with the cap still on), there may be a little bit of fizzing inside the bottle, but the continued pressure inside the bottle forces the carbon dioxide back into the water.
You have a higher pressure of CO2 gas in the air vs in the liquid. CO2 will dissolve in water so the two will equalize so that the pressures of CO2 are the same on each side. If you want bubbly water you just turn up the CO2! Or naturally you seal the container so the whatever is causing CO2 to form increase the pressure. Yeast does this in beer.
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