The gas is carbon dioxide (CO2) and is both dissolved and chemically bound in the water. This is reversible: add more gas (pressure), then more gets into the water; remove gas and some will get out to replace the missing one.
This exchange is relatively slow, so it does not happen instantly. A bottle left open will slowly try to equalize the gas content with the atmosphere. But the atmosphere contains only very little CO2, so effectively all of it gets out given enough time. If you shake the water, this process is fastened a lot because the gas gets more chances to do so.
If you let a heavily shaken carbonated drink sit for some hours, the gas will get back into the liquid and it won’t all spill out when opened. Similarly those devices that carbonate water just push it in with high pressure, so it gets dissolved fast.
An interesting titbit is that this process depends on the _partial pressure_: the pressure exerted by the CO2 only, the other chemicals don’t matter. That’s why “fizz-keepers” are mostly pointless or slightly dangerous. They pump in normal air, which has only very little CO2, so the CO2 from the water still gets out until the CO2 itself is responsible for enough counter-pressure. But that adds to the already present high pressure from the fizz-keeper, potentially popping the entire bottle.
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