Why are the Trappist monks vegetarians?

I was reading about American prisoner Clayton Fountain and then stumbled upon the Trappists who are vegetarians. Generally, as a Baptist and having many Catholic friends, I don't see any vegetarian amongst them unless of course some of them avoid something for taste or health like mutton or duck. I see quite often people from some other religion avoid a certain meat or meats altogether which has their religious basis. I would like to know about the religious basis on which the Trappist monks avoid eating meat.