First Law
The reason why the nozzle becomes cold so quickly is related to the first law of thermodynamics -- the change in the internal energy of a system is equal to the quantity of heat it gained plus the work done on it. You can observe this in effect if you've ever used a bicycle pump, as when you pump very rapidly, the device starts to warm up, even though no heat has been transferred to the system. By pushing on the pump handle, you did work on the air inside, increasing its internal energy.
Carbon Dioxide Extinguisher
The carbon dioxide inside your extinguisher is at an extremely high pressure. When you fire the extinguisher, you allow the gas to escape through the nozzle, and as it flows out it expands rapidly, doing work on the surrounding atmosphere in the process. As it does work its internal energy decreases, so it becomes colder. The difference in pressure between the inside of the extinguisher and the surrounding atmosphere is high, so the cooling effect is quite striking.
Adiabatic Process
An adiabatic process is one where no heat energy is transferred between the system and its environment. The expansion of the gas as it escapes from the extinguisher is so rapid that you can approximate the results by describing this process as adiabatic, even though in fact heat is transferred between the environment and the gas as the gas warms up. For an adiabatic process, the amount of work done by the gas is equal to the change in its internal energy.
Metal Horn
It's not just the carbon dioxide that becomes cooler -- it quickly cools its surroundings as well. Molecules in the surrounding air and atoms of the metal in the horn collide with molecules of the cold carbon dioxide, transferring energy to it and causing it to gradually warm up. The metal in the nozzle is an excellent conductor of heat, so it can rapidly transfer heat to the carbon dioxide. That's why the metal horn feels so cold -- the carbon dioxide is cold and the metal is rapidly losing heat to it.