A great way to show you why exactly a person would be Vegan or vegetarian is to watch the documentary Vegucated. You can watch it on Netflix even. Best way to explain it to someone is that documentary I have found.
Me personally, I absolutely love steak, chicken etc you name it. However the way in which the livestock we eat is punished and treated isn't humane and honestly makes me cringe. I have also found that when filling myself with plant based and whole grain meals I don't have the same crappy feeling I used to with a big steak. Other things such as the correlation between red meat consumption and cardiovascular disease and cancer were things I would prefer to try to avoid.