There are three types of carburetor choke thermostats. Integral, divorced and electric. Integral are mounted on the carburetor, divorced are mounted in the manifold & electric can be mounted to the carburetor, or the intake.
When the thermostat coil is cold, it closes the choke valve. When the engine starts, the air entering the carburetor will open the valve slightly. The vacuum piston will also pull to open the choke valve a little more. The two working together will work against the thermostat coil to let air enter the intake. Otherwise the closed choke valve will not allow enough air to enter the intake and the engine will die.
With the exception of the electric thermostat, most are heated using some kind of choke stove. A pipe connects the exhaust manifold with the choke thermostat either at the center of the choke thermostat, or at the side of the thermostat housing and with the help of vacuum, hot air is brought into the thermostat. As the engine heats up the air causes the thermostat coil to expand and open the choke valve.
Choke stoves are often eaten up by years of not being maintained and parts are usually hard to come by. In these cases a good solution may be to convert to an electric choke. The electric choke doesn't use hot air to move the thermostat so the stove pipe will no longer be needed.