Why is Cobalt II Chloride (CoCl2) often used in hygrometers?

Because the mixture of the used chemicals on the paper strip (cobalt chloride, sodium chloride, calcium chloride, and water) changes color at certain humidity levels.
Note that this method only provides a rough indication of the humidity in the air.

A more accurate hygrometer is made of a hair that stretches - or a piece of catgut that twists - related to the humidity in the air.

The most accurate instrument is an electronic hygrometer (used in museums and art galleries).