A warm front brings warmer, lighter air into a region of colder, heavier air. As the warm air arrives, it encounters the colder air at the surface and slowly rises over it. As it rises, this warm air cools and the gas-state water stored inside it starts to condense into liquid water. Remember that condensation (the change from gas-state water to liquid water) is the result of cooling. In this case, the condensing moisture makes clouds, since clouds are basically nothing more than tiny liquid water droplets.
The clouds produced by warm fronts tend to be even, layered clouds, rather than towering clouds. This is because the rising air, in this case, is gently and gradually rising above the cooler air at the surface. Warm fronts don't charge into a region like cold fronts often do and there is no pronounced wedge of colder air forcing great volumes of fast-rising air (which would tend to produce taller, choppier clouds).
It makes sense, then, that the precipitation caused by warm fronts is generally less intense than with cold fronts. While a more significant event like a thunderstorm is possible, warm fronts are most often associated with light rain and drizzle, often occurring over many hours.