This is not a technical proof, but I think it will help you to understand it better:

Imagine that there are 2m+1 people and every one has a vote. He can use this vote to add 1 point for himself, or to take 1 point off another. As we already know that there will be a winner X (he has at least m supporters). Consider the toughest situation for X to win the election: All the other m people ally with each other and forms one big group. Even in this situation, X will have at least 1 points in the end, while others all sacrifices. Not to mention that the other m people may form smaller groups and fight with each other (Moore Voting Algorithm actually is a reflection of this situation)