We all know how it goes. Music is a part of video games throughout everything, from those artists who exclusively focus on video game music, to those who just want better promotion, the songs and music of video games is what makes them work.
Imagine blasting through Sonic the Hedgehog's Green Hill Zone without the iconic theme. Same goes for the original Super Mario Bros. We may not think about it when actually playing the game, especially during intense moments of play, but when we have time in the game to just sit back and enjoy the music, you can understand why it's there.
It could be for many reasons. To create atmosphere in the level, to link the characters with the stage, or just for the plain reason of having something there and not just sound effects. There are many games that do this, most common of which are racing games. In fact, with racing games, it has got to the point where it's not just music in the background, but to a certain extent, you can control the music that plays.
Forza Horizon - change radio station by using the D-Pad. Need For Speed Most Wanted  or Burnout Paradise - access the menu and select which songs you want from the selection. Or there's the way Gran Turismo 5 does it. Use your own playlists on the console itself. Arcade racers like Mario Kart obviously wouldn't do this, as the music gives their tracks personality.
And then, with the likes of Youtube, we can access the songs away from the game. As I type this, I am listening to a custom playlist of Sonic songs varying between the Adventure series, Shadow the Hedgehog and Sonic Heroes. You can also get CD's full of songs from games in Limited Edition collections, or as standalones, in certain countries. If there wasn't music in video games, I expect we wouldn't have such an iconic fanbase for the games of some series.