At it's highest level, the giving of a thumbs' up or a thumbs' down is simply a way of participating and showing whether you agree or disagree with an answer.
But if you get a Best Answer, you will also get points added for all the thumbs' up - so if you want to be really small-minded, mean and nasty, then you dish out thumbs' down to detract from the points that would be awarded to any answer you dislike or disagree with.
When a particular religious denomination give my answers a barrage of thumbs' down, I console myself in the knowledge that I must have struck a nerve, and perhaps I'm doing something right (lol)!
Edit: As things stand right now, if you were to award me best answer, I'd get 13 points (10 for b.a. and 3 for thumbs' up). So if you bear a grudge against an individual or a particular point of view, you can get into a thumb frenzy in a pathetic attempt to rob people of points. Sad, yes? I only award a thumbs' down when an answer is spiteful or vindictive. However, having read Sculptor's answer, I think I will mend my ways and adopt his gracious attitude. Have a thumbs' up from me!