There are a number of reasons that vegetarianism is associated with yoga. One of the most basic reasons is that yoga developed in India where there is a longstanding cultural habit of avoiding eating meat, particularly beef. This practice is associated with the Hindu faith, but like most things it is a product of a great many cultural, historical, economic, and ecological factors. There was a book written back in the 1970s called Cows, Pigs, Wars, and Witches, by Marvin Harris. I remember reading an excerpt from that book (Mother Cow) talking about how the devotion to cows as a symbol of female fecundity and life itself, and the refusal to eat them was actually a highly pragmatic practice that allowed poor families to have a source of fuel (slow burning cow manure as a perfect fuel to cook slowly while performing other duties), building materials (the cow patties could also be spread out to cover a dirt floor and keep dust out of a building), milk (even from a very underfed animal a little milk could help the protein intake of an equally struggling family, a factory for producing make animals to pull the plows (as opposed to having to buy an expensive tractor), and so on. Of course the economies of India, even rural India, have changed significantly. But in the world in which yoga developed there were many reasons, both practical and philosophical for vegetarianism. The same can be said for our world, and for the modern pull in yoga toward non meat eating.
Most of the arguments toward vegetarianism seem to stem from three areas: health, ecology, and ethics. (A good debater with a particular argument they want to win might bring forth quite a lot of points. I really do not want to judge the rightness of meat eating, but just where I see the connection to yoga. I do have a point of view, and I will let you know what it is, but as in most things I think people need to make their own choices..... but those choices should be based on knowledge, open minded exploration, and ethical rigor over what is easy and quick). All three of these considerations arise in yoga.
I need to leave aside those who take up yoga because it is popular, or cool. I also need to leave aside those who do it to 'sculpt' their bodies. In a way, the quest for the body beautiful is not as big a thing in yoga, as in weight training, and in cardio training, as the end points of those disciplines have a great relationship with how we look. Some forms of yoga are more sculpting, but I think most yoga is done for MORE than for just that.
But in a larger sense sculpting of the physique is part of a concern for physical health, and yoga does effect various parameters of physical health. While the story goes that early yoga was done to strengthen the body to sit and meditate and become more spiritually adept, in fact the idea of gaining health and strength through yoga is pretty long standing. Before India became a colony of Great Britain many of its warriors were practitioners. And if you read old yoga manuals there are many claims made for the health benefits (not with any science to back them up, but of course the use of science to measure the effects of physical practice is no place near as old as yoga is). We know now that physical exercise and a healthy diet are synergistic in the production of health. That idea was also long standing in practitioners of yoga. I would not argue for the health of a completely vegetarian diet for all people, but it is fairly clear that more beans, grains, fruits and vegetables in ones diet are associated with the reduction in quite a few diseases (including various cancers, heart disease, type II diabetes, etc).
The ecological aspects of the yoga/vegetarian association stems from the understanding underpinning the practice that a person is more than one thing. A person is more than just a body, or just a mind, or just a heart.... We sometimes talk about the different 'paths' of yoga as though one needs to choose one: I walk the path of Hatha so I do vinyasa on my mat, you walk the path of Karma so you feed the homeless, someone else walks the path of Bhakti and maybe spends all their time chanting about Devi. Once you come to the mat and begin to explore the self you begin to understand that as you open the muscles you also open the heart. The practice of love and respect must extend beyond the self. A person can spend their life going off to spiritual retreats to discover themselves but never giving to others, or opening to their world will never discover some of the most important things about who they are. I have heart that cattle consume 16 times as much energy as does grain. Livestock is very carbon intensive. So for someone who is opening their heart, who is stretching more than their hamstrings they can choose (to paraphrase Dumbledore ) to go on doing what is easy, or what is good for everyone on the planet, including their own children, and grandchildren. Again, it is not necessary to go completely meatless, but every small act of connection to others in the world is an act of both Bhakti, and Karma yoga.
The ethical underpinning of a vegetarian lifestyle in yoga relates back to Patanjali and the Yoga Sturas. The practice of Ahimsa (non violence) is a core part of yoga philosophy. Also the idea of karma. I am remembering Sir Paul Mccartney's comment that if slaughterhouses had glass walls everyone would be a vegetarian. Some people travel quite far on this line all the way to being vegan. Others simply refuse to support factory farming and eat meat, but only when it is raised humanely.
In my house we represent all of these threads. My older teen gave up red meat some time ago. He still eats fish and chicken (and occasionally pork). His motivator is social action, and concern for doing good in the world. My younger teen, following a conversation on some related topic googled some video of what happens in an industrial pig farm, (the only meat he really eats is bacon or turkey bacon) will eat ONLY meat that is clearly labelled cruelty free. His is the response to stand up, not against eating meat, but the brutality of how living creatures are treated. For me: I have been vegetarian, and I have been a meat eater. I buy meat and cook it for my family, and eat it myself, but I buy less, eat less, cook and eat more beans, buy cruelty free, fair trade, and additive free as much as I can. I find that I shop more mindfully these days.... thinking carefully before anything goes into my cart. Some days I buy potato chips.... and if someone invited me to dinner and gave me steak I would eat it understanding that the joy of being together and sharing a meal is part of living within the 'beloved community' as well. I think there are a lot of ways we can decide how to think about this.... but if we are going to be committed practitioners of yoga we must really think about it.