Well, basically the idea is so people can vote you up or down so that people can see how many people agree or disagree with the answers given. It's an anonymous way to agree or disagree, because actually commenting on someone's answer will display your name, so up or down is a way to do it anonymously. Though to be honest, a lot of people use it as ways to just be petty, and cut someone down. Still, it is good to see a bunch of people cutting down someone particularly stupid, or offensive, and it's nice to see people with good advice get voted up.
Oh, also, at one point, you used to get Xper points when people voted you up, and lose points when you were voted down. That was apparently removed because they felt it wasn't very effective, and some people were abusing it. I can understand that. Especially considering the points are directly tied to rewards that could cost the site money.
But yeah, that's what the ups and downs are for. Agreeing and disagreeing anonymously.
Votes up means people agree and like your answer. Votes down is the opposite. It's just to help the question asker see what most of the users on here agree and disagree with. And get the attention of other users when they read people's answers before providing their own answer. Users on Yahoo do the same in questions and articles.
So that the bandwagon can vote the way of the majority. I have seen many questions here with loads of votes to either side that they never should have received. Answers are all a matter of opinion and should be interpreted as such, but this site is insane with immature bandwagon users.