They believe that eating the fish you catch is just plain wrong. I think that attitude is elitist, short-sighted, and a subject better left for the fishing forum. The practice leads to bigger fish. If you put a fish back instead of keeping it, it gets bigger.
Bigger fish produce better progeny.
And fish that survive in the wild produce fish that will have those same survival characteristics. There's nothing wrong with keeping a fish or two for the frying pan, but if you keep too many if too many people keep too many, and if too many people keep too many big ones the quality of the fishery declines eventually. And with the world's fisheries getting more and more pressure from the growth of sport fishing, it makes sense for us to keep their future health in mind. Catching and handling fish can build up their lactic acid, increase their body's pH rise, and tire them out. Once you tire out a fish and release it improperly into the water, that fish can get caught in the current and get swept away.