Vegans don't eat, use or wear anything that is made/comes from animals :3 So no eggs, dairy, fish, meat, honey, leather, etc. :3
Vegetarians don't eat meat of fish, nor wear leather, etc, but eat eggs, milk, cheese, etc :3
I can't answer for everyone, but a lot do it because it causes the least amount of damage. Yes, it is still killing something, but a plant is not a sentinent being, compared to say, a pig. If you pull a leaf off a plant, it doesn't react. You pull a leg of a pig and it'll move away, scream, etc.
So, to me at least, its about trying cause as little pain and damage as possible
And I don't agree with the way the animals are treated these days.
So for me, it's not so much eating meat that I'm against, but rather how the animals are treated before hand. Personally I believe that just because you're going to eat something doesn't mean you shouldn't treat it with respect.
Some others do it out of love for animals, some are repulsed by the idea of eating another living being, some people don't agree with the methods, some for health reasons :3 There are a lot of different reasons that people are vegetarian or vegan :3 Depends on each individual person ^^
I have pretty bad asthma too- but it's calmed down a lot since I cut dairy. I've heard that goats milk does the same, but I personally think it tastes disgusting XD
Rice milk is lovely, and some soya milk is ok, but it depends on what you like I guess ^^