They don't.
I couldn't tell you why some people who eat fish claim to be a vegetarian, but they are incorrect. I'd imagine that it is the mistaken belief that fish don't feel pain, or that fish aren't treated badly prior to being killed, or something like that.
Fish are like any other animal. They feel pain in a meaningful way, have memories and can be trained, and demonstrate an understanding of their community and surroundings. Fish are often farmed, which is pretty gross and cruel to them.
Trawling is horrible for the environment and extremely wasteful. The nets drag on the bottom of the ocean, stirring up mud and wrecking the plants and animals that live down there (coral, anemones, etc). Anything that isn't part of the target catch is tossed back overboard, either dead or dying. It can be up to 80 or 90% of what a boat brings in. That would be like killing ten chickens and only using two (while throwing the rest into the garbage).
In short, I have no clue why people would think that it's wrong to kill, say, a chicken, but not a fish.