As the water is being drawn across the fill, it starts to evaporate and also atomize (meaning that the streams of water break up into tiny droplets that are technically still liquid, but are light enough to be carried away in the moving air stream). As these water droplets are pulled into the outside air, they can be carried anywhere. Often, cooling towers are located on the roof of buildings. The other thing that you'll often see on the roof is the building exhaust fans and the fresh air make up fans. If the fresh air makeup fan inlets are located anywhere near the cooling tower, it is very possible to have those same tiny water droplets get sucked into the intake, and pumped into the building along with the fresh air makeup.
Mechanical Engineers usually design the location of these intakes to be far enough away form the Cooling Towers to prevent infiltration, but wind currents can be a little hard to predict. Also, if the Cooling Tower isn't being operated correctly, there can be more water atomization than there should be. For example, if the Variable Frequency Drives (VFDs) that control the Cooling Tower Fan speed isn't set up right, it can run too fast and pull out more water droplets than it should be (this should ordinarily be kept to a minimum because makeup water isn't cheap, and it's not "green" to use too much water).
Hope that helps.