Quote from the blue box on the right:
"Upvote answers or questions that have helped you."
"If you get a good answer, click the checkbox on the left to select it as the best answer."
It honors the author of the answer (or question) and rewards him with points.
Downvotes will decrease the points a bit and should only be used, if you really dislike the answer or question. e.g. if the answer makes no sense or is not related to the question.
