There is no universal answer to that question, as it differs between contest/teams.
In cup games, especially in international ones, the award is often handed out by sponsors of the tournament/cup. This was also the case when the MVP of the entire World Cup was to be chosen. This was done by the German sporting brand Adidas, who's one of the main sponsors of the WC.
[Trivia: there were some speculations in media as to whether or not Leo Messi was awarded MVP simply because he has a sponsorship deal with Adidas already, but this has not been verified].
In leauge games, the sponsors, (local) media or fans of the home team is usually deciding who's to be awarded the man of the match. The award can simply be honor and gratitude, but it can also be monetary or constituted by some sort of perk. For example, a couple of years ago, the man of the match in home games for the Swedish team Elfsborg was given the keys to a Porsche painted in the Elfsborg colors. The player was then allowed to drive the car until the next home game, when a new man of the match was decided. This is a typical example of a PR-trick as well as a carrot for the players, which can typically be combined with the award.