Water can dissolve a certain (low) quantity of CO2. To create this "refreshing" effect (that is when all the nerves in your gullet fire because they have the impression to being dissolved by an acid :-)), a lot more CO2 is dissolved in the soda.
But the liquid can't keep so much CO2. To keep the CO2 there for a certain time (say a couple of months), the bottle is under pressure. This way, the CO2 molecules will spread evenly in the liquid. That only works because the liquid can dissolve some CO2 in the first place. It wouldn't work for something that doesn't dissolve (rock, oil, ...).
A liquid can't be compressed, so the volume of the water doesn't change when you open the bottle. You will get a little bit of fizz from the small amount of air on top of the liquid. But most of it will come from the CO2 which isn't kept in the liquid by the pressure.
Why does CO2 dissolve? It actually forms a (very weak) molecular bond with the water. This way, all the molecules can reach a lower energy state (the water forms "pouches" for the CO2 molecules and they "rest" in there unless disturbed).
This is very fragile because the water doesn't really want to dissolve so much CO2 -- it just has to because of the pressure under which this happens. Now when you share the bottle, you disturb this fragile balance. The CO2 molecules move and start to form tiny gas clouds that tend to absorb each other (because they like each other more than the water): You get bubbles.
If you don't open the bottle and wait for a few days, the water will dissolve the CO2 again. This is slow for many reasons. First of all, the water doesn't really want to. Second, no one mixes the water, so the part near the surface has a high saturation and that disperses very slowly through an undisturbed liquid. Mixing the water would be bad, even.
In the production plant, a nozzle is inserted into the bottle and the CO2 is released near the center of the liquid. If the bottle wasn't sealed, the CO2 would quickly evaporate. But since it is sealed, the CO2 spreads evenly in the liquid (so actually the water dissolves the CO2 bubbles, not the CO2 itself).
This is the reason why you only get a small fizz when you buy a bottle since it has been shaken a lot before it arrived at your place (moving to a storage, on a train/truck, loading/unloading, stacking onto a shelve, carrying it home, etc).