Well, it's quite simple. Just look at answering questions as a kind of "competition" between answerers. If the same question has several answer, the answerers are competing among each other, and the answer with most upvotes is on top. (Well, usually - it depends on how user chooses to sort the answers, but this seems to be the most frequently used setting.)
So if your main objective would be to maximize reputation, it's better for you to have your answer among the higher voted answers near to the top - since it is more visible. But if other answers are good, it is nice of you to upvote them - even if this means that your answer might get less visible. So in this way your behavior is "sportmanlike" - you help people who "compete" against you, in a sense.
(Whether or not looking at participation on MathOverflow and other Stack Exchange sites is more useful or more harmful is up for debate - probably it has both positive and negative aspects. I have mentioned this here mainly as an analogy to explain the choice of the name. Regardless of whether you consider gamification useful or not, this badges rewards behavior which I would definitely consider as a positive contribution.)
Does it mean that the winner has asked 100 different questions each question receive at least one +1 and each one has at least one answer with at least one +1?
No. It means the you need to have 100 answers (with positive score) and in each of those instances you have to upvote at least one of the competing answer.
Users who do not have the badge can check progress towards it in the usual way - using the badge tracker on their profile page.