There could be different reasons for it, but most people probably are doing it for their health and not out of any ideological reason. I know it was for me. Money saving factor could also influence some people.
Anyway, from what I read (correct me if I'm wrong) Americans on average eat much more meat than we do here in Europe, some probably 3 times a day, with every meal.
We usually ate meat once a day in the evening, with dinner, for the rest it was mostly cheese/dairy and eggs, with cold cuts sometimes, so it made the switch much easier for me.
The problem is, you can't just cut a major source of protein out of your diet and expect that nothing bad happens. I guess we all know horror stories about vegans losing their hair and getting quite sick in the process. I'm not a big fan of veganism in general, though some of my readers pointed out the diet worked for them. I guess it could for some people, especially middle aged men with heart and circulation problems.
This has been a subject of many heated discussions, but as somebody pointed out, there exist cultures which have been vegetarian for many generations, but none has been vegan. In my opinion, you should at least eat some dairy and/or may be fish, because we need at least some amount of animal fat (not so much animal protein) in our diet.
Anyway, if you cut out meat you will need to substitute it with something. You could start by eating more chicken and fish. The current guidelines in the Netherlands is to eat fish at least once a week, but the American recommendation is to eat it at least twice and better, 3 times a week. That would give you 3 meatless dinners already.
Health benefits of fish are well-known, just to add anther one: apparently, researchers in UK found out that women who ate more fish had a delayed menopause, which means that their fertility lasted longer.
If you decide for a more vegetarian approach, you can substitute one meat portion (3-4oz) with 2 eggs or 70g of regular cheese or with higher amounts of low fat cheeses like vegetarian mozzarella (6g fat per 100g) or cottage cheese or skyr. The important thing is to look at protein amounts. Many factory made meat substitutes have very low protein content, which makes them rather inadequate. Cottage cheese, on the other hand contains 26g of protein per 200 g with very low fat content. However, it's also low in zinc.
You should take care to get enough B12 (fish, dairy products, eggs) and zinc, since fish doesn't have much of it, unlike some other sorts of seafood.
I'd also like to say a few words about soy. Soy is a very controversial food, supposedly it lowers testosterone in men and makes them effeminate (some researchers deny it), then there is a discussion whether it's good or bad for breast cancer (inconclusive), some of it is genetically modified and it can also lower your thyroid function. It can also suppress your ovulation and disrupt your cycle, so if you are a woman of child-bearing age, it's probably wise to avoid it, or only consume in small amounts.
Many people switch to (semi) vegetarian diets to prevent heart disease and hypertension, and it's generally recommended that such people follow a low-fat diet.
This type of diets aren't well-suited for non-menopausal women, unless there are very fat, of course, and need to lose weight. If you are a normal weight and stop eating meat and poultry for whatever reason, you'll need some butter and full fat dairy in your diet, since it's been shown by research that low fat dairy interferes with ovulation (again). That's why so many vegan women become (sub) fertile, they miss animal fat in their diet.
In general, if you are under 30 and normal weight, you can probably eat anything and stay healthy. And women are protected from heart disease generally, as long as they menstruate and don't use the pill (which raises the chance of blood-clotting, btw). After they go into menopause the risk starts increasing and catches up with men at some point. At this age, soy probably won't hurt you any more.
So that are just some tips I've learned. As of now I'm following a pescatarian diet. I'm not losing weight this time as I take care to eat more eggs, butter and fat, and eat fish 3 times a week. I'm not ideological about it, so will probably eat meat in future, in any case with Christmas as I'm not planning to cook 2 Christmas dinners.
And yes, quitting red meat did improve my health in some ways, so that's the reason for it.