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The simply answer is that your A/C dumps heat outside using big metal fins, it's not terribly great, but it works well at that scale.
Dissipating it into the air for the amount of heat some data centers need to get rid of doesn't cut it, so they use evaporative coolers.
The phase change of evaporating water from liquid to gas uses approximately 7x more heat energy than taking room temperature water and getting it up the boiling point itself.
Essentially they stick their large metal fins from the AC into a large pool of water and boil it off. This gets rid of the energy with a much smaller and cheaper system, but uses up water.
Edit: To clarify on the water in your home AC, the water is actually being collected INSIDE on the chilling unit as lowering the temperature decreases the ability for the air to hold water, it's then being pumped outside. Data centers recirculate most of their air, letting in only a small amount of fresh air that they pre-chill, rather than letting fresh air in from the outside all the time like your home does.
Home hvac units donโt let in fresh air all of the time, they recycle air.
The HVAC system no, the home itself, yes.
Depending on how old your home is of course, newer homes tend to have lower exchange rates.
Also datacenters don't have windows, or even doors constantly letting people in and out of cooled areas and outside.