That "empty" space isn't actually empty. It's filled with carbon dioxide gas, the same thing that creates the carbonation effect in the soda water.
Carbonation is what we call the dissolution of carbonic acid in water. Carbonic acid is basically just a molecule of carbon dioxide and a molecule of water, rearranged into a single molecule. This isn't a very stable molecule and even the ambient energy of the solution is enough to break the carbonic acid into water and CO2. The CO2 prefers to be a gas at these temperatures, and so it bubbles out of the solution. This is why you continually see bubbles forming in a carbonated drink, until the carbonic acid is totally depleted and the drink goes "flat."
When the drink is inside the bottle, several factors are at play. The carbonic acid "wants" to break apart, but if the carbon dioxide gas has nowhere to go, then it will dissolve back into the water again. However, if the gas can escape the liquid in any way, it will do so, until the place it has escaped to is totally saturated with the gas. For example, when the drink has been opened, the place it's escaping to is the atmosphere, and the atmosphere has a much bigger capacity for carbon dioxide than the drink does.
So, if there's any possibility for the gas to escape, it will do so. However, by giving it a place to escape to, but then completely saturating that space with CO2, the gas will simply cycle back into the drink, retaining the original level of carbonation. This gas is pressurized to make sure that it is able to "force" the CO2 back into the drink, which is why the cap hisses when you release it.
If the gas wasn't injected into the top of the bottle, the first thing you'd encounter on opening the bottle would be highly-pressurized soda water that would immediately begin leaking out of the bottle.