So many different reasons, of course, with countless books and blogs having been written on the subject. Common reasons given include "closeness to nature," relaxation, food, sport, comradeship and a psychological "calling" akin to a primordial hunting instinct, the last being a bit questionable, in my view.
There is, I think, a more fundamental reason. It is knowing something very deeply, not just intellectually but experientially. The intersection of many things at an instant: the trout's behaviour, the structure of the river and its currents, the shape of the river bed, the weather, the fly life, the expert use of rod and fly-line, the tying and presentation of flies, your own limitations, the sound of your own heart-beat, the structure of your own consciousness.
Fly fishing, in its deepest sense, is a phenomenological inquiry. Hence the need for restraint as well put at Classical Angler .