Why? First of all, I was taught to appreciate it at a tender age, some 60+ years ago, by a gruff great uncle (with a heart of gold!) Under his watchful eye, I really learned to ENJOY myself. Yes, I tried the other methods as they came along, but never with the satisfaction I enjoy from fly fishing.
Why do I persist in deer hunting with an old traditional flintlock? Why do I hunt grouse with a muzzleloading shotgun?
Maybe it's the gear - most of which you can make (at least partially), or the excitement of a rise, or seeing your indicator disappear. The peace and serenity of the surroundings has a lot to do with it, I am sure.
Because I enjoy these things more than anything else!