I avoided all the hype over "The River Runs Through It" movie, and therefore, avoided the book as well. Mistakes do happen!
No, it's not really about fishing. What is it about? I think that is really up to each reader. It's about religion, and greater good. It's about family relationships, and beautiful scenery. It's about being human.
Actually, I read it as a travel book. Not to a place, although there were some wonderful description, but to a time. To the early 1900s when the West, while not exactly wild, wasn't exactly tame either. The first story, which gives the book it's title, is the deepest. But the others give a great sense of what it might have been like to live in the West, when watching for forest fires meant camping just below a peak, and walking two days into base camp from civilization.
(I did not read the edition with the forward by Annie Poulx, and cannot comment on it, but what a perfect author for a forward!)