A cold front brings colder air into a warm region. Since cold air is more dense and heavier than warm air, it remains closer to the ground as it arrives. The warmer, lighter air that's already in place rises. You can think of the cold front as a wedge that slides in at the surface like a shovel and forces the warm air to rise.
Rising air is what makes clouds and precipitation, because as the air rises, it cools and the gas-state water inside of it condenses into liquid water. Remember that clouds are made of tiny liquid water droplets. As soon as you see a cloud, you know air has been lifted somehow and cooled. The tallest, most dramatic clouds and heaviest rain are usually caused by cold fronts, since they are usually the fastest, strongest fronts capable of causing the greatest amount of air to rise the fastest. Thunderstorms are also most often associated with cold fronts.