When the cold air touches the warm air, the humidity in the warm air will condense and fall as rain.
The intensity of the rain will depend on how much humidity is in the air and the temperature difference between the cold and warm front.
If a 30°F cold front touches a 70°F warm front and there's a lot of moisture in the air, then it results in a massive thunderstorm.
If you see the doppler radar pictures on the news, you'll see that rain usually comes in bands. That band of rain is the leading edge of a cold front.