A number of reasons:
1. Health reasons. Some people believe that an animal product/by-product free diet is better for you.
2. Moral reasons. These come in two flavours; negative vegans/vegetarians and positive vegans/vegetarians.
Positive veggies are vegetarian for boycott reasons, they believe that the production methods of these animal products are cruel but could be improved to the point where their production wouldn't be immoral in and of itself. So, they boycott the product until such time as the industry improves its standards.
Negative vegetarians/vegans believe that the exploitation of animals for food and clothing is and would be wrong under any circumstances so they don't partake in it.
3. Spiritual/religious reasons, I suppose? I'm not religious or superstitious in any way, shape or form so I won't comment further on this.
For my own part, I was vegetarian for years and took the vegan plunge a couple of years ago. My reasoning was that if you're vegetarian for moral reasons, it's logically and morally dishonest not to be vegan because you can't have a dairy industry without a meat industry, it doesn't make economic sense, they work hand in glove with each-other. If you consume dairy, you're still supporting the meat industry.
As for the "What would happen to the animals/these animals wouldn't exist" arguments I'd say this. For the animals that are already bred, I don't think any of us, barring the most zealous, unrealistic and naive vegans (read: crazy), believe that the world would turn vegetarian overnight. If it were to happen, it would happen gradually, and the production of these animals would decrease in line with that. As for the animals that would never exist in future, I don't place value on potential life, it doesn't exist so it can't be subject to any significant interests.
Speaking of significant interests, that's my main moral reason for not partaking of animal products as best I can. I don't see any moral difference between humans and non-human animals. We're subject to the principle of significant interest; we can feel pain, fear etc so we should be treated accordingly (for instance, if, as some have suggested, molluscs like oysters and mussels have no more capacity for pain than a cabbage, then morally I'd have no problem eating them).
As for the argument that the production of plants involves the killing of animals too (I don't have any figures on this so I don't know how true or untrue this claim is), I'd probably invoke the doctrine of double effect (google it, there'll be loads of articles on it that could probably explain better than I could). I don't think any system is perfect but you should do the best you can.
Finally, on a personal note, pleeeeease don't lump me in with hippies and PETA. I don't support PETA, I loathe them for their policies and for making my side look bad. Secondly, I don't wear dangly earrings, I'm not obsessed with hemp, I don't believe that crystals have Powers (they're pretty, mind) or believe any other supernatural stuff like that. I hate PETA. I don't hate hippies, I just think they're silly.