As another answer states, it's due to pressure, but here's some more information so maybe you understand better.
Below is a chart which shows how the relationship to pressure affects the boiling point of water (from this website):
In the chart, it is using 14.7psi as air pressure at sea level. As the pressure rises, so does the boiling point. A typical cooling system runs at between 16psi and 22psi (I think that's as high as I've seen, but it doesn't mean they don't run them higher). On the chart, you'd add the pressure of the cooling system (let's use 16psi as the example here). The total pressure of the system would be ~31psi (14.7 for pressure at sea level plus the 16 for the system pressure). That would put the boiling point of water between 250° and 254 °F. This is just water. When you add anti-freeze/boil solutions to the water, it becomes higher than that.
How does the pressure in the system raise? Just like you stated that the steam in the system creates this, but it doesn't compress the coolant itself, because you can't compress liquids. As heat is introduced into the system, the liquid itself expands as does any steam which forms. This creates the pressure and is regulated by the relief valve located in the pressure cap as well as the thermostat/radiator, which keeps it cool enough so the coolant doesn't boil. Depending on the system itself will dictate how pressure is relieved and how it will behave.
The normal operating temperature of a vehicle will rarely ever see 250 °F. Your engine will most likely start seeing a large amount of ping/knock at this temperature. Most modern vehicles run in the 200-210° range. Older vehicles ran cooler down below 195°, but that's a tale for a different day.