Why do people choose to eat a vegetarian diet?

Let me begin by stating that there are different levels of vegetarian diets.  For this answer, I'll provide reasons to be on a 100% vegetarian diet.  
It's healthier.  In some cases this is absolutely true.  In some cases this is not true.  For example, most meats are complete proteins.  This means that they contain all 9 essential amino acids.  A vegetarian can still eat a diet that contains all 9 essential amino acids; however, the person must be much more intentional about mixing and matching foods so that all 9 of those amino acids are consumed. A vegetarian diet that does not contain all 9 of these amino acids might not be healthier than a diet that contains at least some meat. 

For moral reasons.  Some people don't like or don't agree with the killing of animals in order to eat them.  Being a vegetarian means that their personal diet does not contribute to further killing of animals. 
Lower risk of disease.  Vegetarians tend to have a lower risk for things like heart disease, diabetes, high blood pressure, obesity, and stroke.  This is likely because a vegetarian is consuming fewer foods that are high in cholesterol and saturated fats. 
Potential to live a longer life.  Studies exist that show vegetarians typically live longer than non-vegetarians.  
http://veg.ca/go-veg/the-7-day-veggie-challenge/top-ten-reasons-to-go-vegetarian/

https://www.accessdata.fda.gov/scripts/InteractiveNutritionFactsLabel/protein.html

https://www.downtoearth.org/go-veggie/top-10-reasons