I love game stories, because I see so much potential in video games as a story telling medium. In fact, I don't think anyone has really capitalised on just what video games have to offer yet, but people are getting there. I'm a writer myself, and, hence, I read a lot, and I think that's probably what makes me like video games as much as I do; literature and video games are the only media that can truly convey storytelling from different perspectives.
In a video game, and in a book, you often are the protagonist, through use of first-person, but video games have an advantage over literature, in that they're a multi-sensory experience. Video games are at their best when they tell stories without words; when the very actions you take in a game are a seamless part of the story. YOU'RE doing this because YOU need to. Changes in the story affect you, because it was YOUR actions that brought you this far. A truly great game blurs that line between you the player and the game itself.
Then again, I'm also a person who likes to read a lot into things; I like being made to think, and since so much time in video games is spent in 'silence', if you will (that is not being given exposition, not watching cutscenes, not being told what you should be feeling), I find that it gives the imagination ample room to prosper - to take in the experience and unravel it.
Some games are better at this than others, obviously, (in fact, most games have horrible storytelling) and I'm sure there's plenty of people who really couldn't give a shit about this stuff and end up having completely different experiences than I do, but, then, that's part of the beauty of it. Games and your experiences with them reflect something about you as a person, because you're in control. Your actions in and reactions to a game are a manifestation of your mindset, which is just so deliciously Freudian.