Mostly, it's an aesthetic thing so people can see how much you've played the game. You gain experience points towards your player level for pretty much everything that you do while playing. Even losing a match grants you experience points, although not as many as winning a match.
Almost everything in the game is also unlockable simply through having a specific player level. There are alternate requirements for some things, but the developers wanted to make sure that even people who didn't go out of their way to specifically unlock some characters would be able to get everything. If you beat the story mode and just play a lot in whatever other mode you enjoy, you'll eventually have access to everything the game has to offer.
It has no direct impact on gameplay, except that you may not be able to use certain characters until you unlock them.