This started by accident. Barry Gibb tried out a falsetto voice to provide the backing for an early Bee Gees track, Nights on Broadway in 1975. The producer of the record asked the band to do some 'screaming' to make the backing a bit more exciting. Barry started singing higher and higher (he had apparently never tried this before) and realised he could do it easily.The brothers were surprised to find how good it sounded so then they all tried it and realised that they were good at falsetto and it gave them a very different sound. Nights on Broadway was a big hit, their first for some time, and they used the same falsetto voice for their next album, Children of the World. By now it was established that the public loved their distinctive sound and other features of their music, so basically they stuck to it from then on.
Perhaps the most famous example of the Bee Gees' use of falsetto is in Saturday Night Fever, for which they created and produced the music a few years later.