You may have noticed that on a cloudy day you can still see things. The sky is not totally dark, like it is at night. This shows us that some light is getting through the clouds. That light includes ultraviolet light that can cause sunburn and skin cancer.
Cloudy days are usually cooler, so the skin does not feel warm, and there is less of an urge to get into the shade or go indoors to cool off. So you might actually stay outdoors for a longer period. If you are at the beach on a cloudy day, you may be wearing less clothing, allowing more light to get to your skin.
Also, on cloudy days, you may not remember to use sunscreen.