Climate change is moving water around, not creating or destroying it. Warmer air holds more water, so overall, the atmosphere can hold (and at times drop) more water than before. Permanent ice is melting as well, so that puts a bit more fresh water into the air and ocean. The water in the atmosphere is constantly circling the globe, forced largely by the rotation of the earth. Warmer temperature also makes for more evapotranspiration, so more fresh surface water is pulled into the air. But that same water will eventually fall elsewhere.
The sun is the source of energy that drives wind, rain, and evaporation. When you trap more of that energy with GHGs, it just turns up the volume for all of those things. There’s always seasonal and geographic variability, but the extremes increase because all those phenomena are solar powered.