It's a tradition that is centuries old. Fly fishing is not normally the most efficient method of fishing, so maybe the periods of not catching fish lead to musing about the activity. Then, with the thought processes already going, when success comes it leads to other musings about why. Whatever the reasons, fly fishers write about their avocation. I can think of only two who are accomplished painters but many who are writers, and some of these writers are well known and respected literary figures.
I'm thinking about fly fishing and literature because of a book, Teaching Trout to Talk- the zen of small stream fly fishing, by Stuart Bartow, a professor of writing and literature at State University of New York. In 172 pages Bartow includes quotations from the following:
He also quotes from William Humphrey, Rick Bass, and Richard Brautigan who all wrote fishing pieces as well as novels. Then there are Harold Blasdell, George Harvey, and Izaak Walton who are known only for their fishing writings, as far as I know. And finally, Bartow quotes from The Odyssey, The Old Testament, Dr. Suess, and a number of Asian Buddhist writers and poets.
With all those literary references, one may wonder if the author has room left to include much about actual fishing, but he does and what he has to say is honest and accurate.