Clouds form in areas where the relative humidity exceeds 100%. Add enough dry air to a cloud and the relative humidity drops below 100%, making the cloud dissipate. It's that simple.
Or almost that simple. Entrainment occurs because clouds are large. The dry air isn't instantaneously injected throughout the cloud. It instead erodes the cloud from the edges. The process of evaporation decreases the temperature thereby increasing the density. This in turn makes the cooled air mass sink, which allows more dry air to enter the cloud. The relative humidity decreases even more as the sinking cooled air warms.
Or perhaps not that simple at all. Dry air can at times create clouds, very nasty clouds, called supercells. In the North American Great Plains, a large mass of very dry air can be flowing eastward from the Rocky Mountains and meet with a large mass of very moist air flowing to the northwest from the Gulf of Mexico. Because dry air is more dense than than moist air, the dry air at the boundary between the two air masses can slip underneath the moist air mass. This buoys the moist air upward, thereby creating clouds. IfThis phenomenon of a dew point difference between the dry and moist air is known as the Dry Line. If conditions are just right (or just wrong), those clouds can become very large and very tall. A supercellSupercell can result in large hail, and at times, tornadoes.