Today we don't hunt for food anymore (oh, sure, some people still hunt animals, but it's more for sport than for any actual need; if you need meat, buying it at a supermarker takes less effort and money than gearing up and going hunt). We raise enormous numbers of animals in captivity, often in brutal conditions, we feed them hormones and other stuff, and finally we chop them up. Unlike other carnivores and omnivores, we don't serve the purpose of keeping the number of preys under control, we don't contribute to natural selection (sometimes we even thwart it by favouring the reproduction of animals that wouldn't be able to survive on their own, like that kind of cows that is unable to give birth without a person pulling the baby calf out), and our funerals are more damaging than good to the environment (must build cemeteries and coffins, and keep the body locked inside a box where it can't serve as plant food).
Personally I'm not a vegetarian, but I can easily see how someone like Diana, who is exceptionally sensitive and grew up surrounded by unspoiled nature, may think "screw this" and give up on Man's World meat.