Generally speaking, clouds act as a "heat shield". Much of the incoming solar radiation from the sun is reflected or absorbed by the clouds. If clouds prevent incoming solar radiation from reaching the surface, then how do temperatures rise during the day on a cloudy day?
1. Clouds do not prevent all of the incoming solar radiation from reaching the surface. Some radiation still manages to penetrate all the way through the clouds and make it to the surface. If no radiation made it to the surface then it would look like it was night outside during the day. This diffuse radiation that manages to make its way to the surface can warm temperatures a little as the day progresses, especially if the sun angle is high and days are long. In winter when sun angles are lower and days shorter, solar energy will not be able to warm the surface much on a cloudy day.
2. The cloud height and thickness are important. High clouds tend to have a lower moisture content than low clouds. This occurs because high clouds are located where temperatures are much colder. Cold air has a low capacity to contain moisture. Thus, more of the sun's radiation is able to penetrate high clouds and make it to the surface as compared to low clouds. Thin clouds such as cirrus will let more radiation pass through as compared to thick clouds such as nimbostratus.
3. Temperature can warm on a cloudy day due to thermal advection. Warm air advection can transport warmer air into the region. Warm air advection often occurs when the wind direction is coming from the tropical latitudes such as in the warm sector of a mid-latitude cyclone.
4. If precipitation is falling from the clouds on a cloudy day, temperatures will not warm as much as when there is no precipitation. Evaporative cooling from the precipitation will partially offset any warming during the day. When some warming does occur, the dewpoint will tend to rise since rain continues to try to saturate the air.