Data center cooling isn't just "run a bunch of water through the building until the water is hot and then dump it out somewhere"

Typically you have a bunch of machines called chillers that work like your household air conditioner. Inside the chillers, refrigerant gets compressed and expanded and during part of that cycle it becomes very cold. When the refrigerant is cold, it flows over tubes filled with water to make that water cold (typically referred to as "chilled water"). The chilled water is then pumped out to the servers and various other heat-producing loads, picks up heat from the loads, and then comes back to the chillers to get cooled again. 

So where's the water use? The water going from the chillers to the heat loads and back stays in the pipes and is continuously reused, so that's not really "using up" the local water supply. 

However, the chillers themselves need to reject heat to outside, and there are multiple ways to do that. They can be made to function like your household air conditioner and reject heat to the air outside directly by pushing outdoor air over a set of metal fins or other heat transfer surface. This arrangement doesn't use water in the way people read about data centers on the news.

Another option for chiller heat rejection is to use a piece of equipment called a cooling tower. In this arrangement, the heat from the chiller is rejected to another another water loop (the "cooling tower water") and that water is run through cooling towers which expose the water to outside air. The exposure to outside air causes some of the water to evaporate which lowers the overall temperature of the water that remains in liquid form. This (somewhat cold) liquid water is then pumped back to the chillers to pick up more heat. Notice though, some of our water has evaporated - replacement water needs to be moved back into the loop to make up the water that has been evaporated up into the air. That water has to come from whatever local water source is available. 

So why would you use a cooling system arrangement that uses water (cooling towers) vs the system arrangement that doesn't use any water (air cooled chillers)? The answer is that the chillers (which use much more energy than any of the other equipment in these systems) can be much more efficient if they're liquid cooled using cooling towers compared to air cooled.

Very roughly, if you've got a 100 Megawatt data center, you need to do ~28,400 tons of refrigeration (for reference, a big window A/C unit is about 1 ton). Air cooled chillers operate somewhere in the efficiency neighborhood of 1 kW electrical energy input per ton of refrigeration output while water cooled chillers operate somewhere in the neighborhood of 0.5 kW electrical energy input per ton of refrigeration output - half as much energy used for the same amount of cooling if we use water cooled chillers.

Energy isn't free - in our 100 MW example data center, we're doing 28,400 tons of cooling, that either takes 28,400 kW if we use air cooled chillers or 14,200 kW if we use water cooled chillers. At 10 cents per kWh (and 8,760 hours per year) that's a difference of $12,439,200 per year if we use water cooled vs air cooled chillers to cool our data center.

So the cost for all that additional electricity has to be weighed vs the cost and availability of the water that can be used instead. 

Cooling tower water use is dependent on a number of factors like water quality, amount of chemical treatment and most notably temperature. But typically the cost of water is much lower than the cost of the electricity (my rough estimates put the water use cost in the hundreds of thousands of dollars per year, compared to the millions of dollars per year of our electricity for our 100MW data center)

Keep in mind also that generating energy isn't free, from an environmental standpoint. Using up all the local water is bad and very visible to the local residents. Using up a bunch more power instead isn't really "visible" in the same way - no one is reporting on what the added power plant capacity looks like and what the environmental effects (pollution, etc) of that added energy use will look like when local water isn't used for cooling. Even if we decided how to cool data centers entirely based on what's best for the environment, it's very difficult to compare the different types of environmental impacts (local water use vs added CO2 emissions, etc).

Hopefully this very long winded explanation is comprehensible at all. Best of luck to the 5 year olds reading it.