As the warm air rises up, the temperature falls down (that is, the air cools down) and reaches the dew point. At this stage (when the air temperature is equal to the dew point), condensation starts. Condensation is the process by which water converts from gas phase to liquid phase, that is, from water vapor to liquid water. An example of condensation is what happens to the outside surface of a glass of cold water, when left out in the open (we see small water droplets on the glass surface).
Condensation of warm air results in formation of clouds, which will ultimately cause precipitation. This process is a key component of the hydrologic cycle or the water cycle. After precipitation, water comes back to the surface of Earth and will evaporate at some point to time and thus, will complete the cycle.
Hope this helps.