In the series, Avatar Kyoshi used them to augment her bending, and in turn, the Kyoshi Warriors use them as a way to mimic Kyoshi, as well as to strengthen their technique of redirecting energy.
Some styles of martial arts, in actuality, use fans as offensive and defensive weapons. Tessenjutsu, for instance, is an art which uses the Japanese iron fan tessen, an elite weapon that could be used to defend against projectiles and weapons, as well as provide offensive force when needed by the wielder. These fans were made of a heavy metal, similar to the fans Kyoshi and her warriors used, possibly hinting that the basis for the Kyoshi Warriors' choice of weaponry originate from Tessenjutsu.
The further use of make up and fans could relate to the Geishas of Japan, entertainers in dance and arts. By using Tessenjutsu as a basis for weaponry, and Geishas as the basis for their appearance, the Kyoshi Warriors are provided with a look that represents both femininity and strength.