Do you mean specifically how do they work or how do they work in the scheme of controlling temperature?
Specifically, a motor controls the damper blades based on a signal, usually from a thermostat.
In the scheme of controlling temperature, they modulate airflow to maintain temperature in a space. They shouldn't be designed so one zone starves the other though. Systems can be dumb, meaning they don't know what other zones are doing or systems can be extremely intelligent, switching over heating/cooling based on demand and settings, incorporating PID loops, resetting static pressure.
In your situation, the system is probably a constant volume system and the zone damper system will be fairly dumb. That means if you want to incorporate zone dampers, you will want to have a bypass to relieve pressure in the duct when your dampers close down. This can bypass directly back to the return duct or dump into the plenum. Either way, you will need a third damper that is either mechanically set to relieve pressure at a certain set point or automatically controlled based off a pressure sensor.
There's more to consider with these dampers, such as insuring that you maintain minimum ventilation rates, what do you do when one room needs heating and the other cooling, how much complexity do you want, etc. There should be some good articles on google.