This article is honestly me cheating a bit as I would have preferred to write a true analysis or something more comprehensive than a question, but I’m busy! So this is what you get. But don’t fret, I think this question is actually extremely interesting, and very important.
Why do we play games? I ask this because I recently got into a debate and one participant countered criticism about a game’s setup with, “I hear people play games for the story.” Now this very well may be true since many games have fun stories, but so do books and movies, and you don’t have to fight your way to the next bit of stories in those. You don’t have to spend hours jumping from one plot point to the next. So why do we turn to games for story when we have books and movies?
To me I think the answer is “participation.” Games allow you to participate in the story. But it is with this answer that I then begin to question certain games which don’t let me actively participate in the story, but instead just force me to do task after task that holds no real meaning in the overall narrative. Along this vein, should we forgive games with great stories for their bad gameplay? I could go on, but I actually wrote a bit about this previously in my article about games and art, but I think we can go further into this question.
Since I need to get going I’ll leave the floor open for you to counter, explain, extrapolate, divulge, or what-have-you in the comments below.
Fistful of Wits 
The trick is, ‘games’ do so many different things, in so many different ways, that I’m not sure your question has any reliable, all encompassing answers.
On the one hand, “why do we play?” is a question with a lot of history behind it, some fascinating answers, and even more follow up questions like “why do we play games with rules?”
On the other hand, the interest in this discussion seems to be “what do we want out of playing a video game?”
There’s a lot to be said for the draw of simple task-reward mechanics, even in entirely story driven games with few of the familiar ‘gamey’ systems like experience points, heads-up-displays or victory conditions. That is to say, it’s those little moments of success, whether it be increasing a number or getting a virtual nod and smile, that help trigger the brain patterns we call ‘fun’. Such moments exist in all but the most far-flung artsy and experimental games. Sometimes it may be as simple as the reward of learning something new, or filling in blank space on a map- speaking of which…
In some types of games, there is the opportunity for exploration. Not just ‘sandbox’ or ‘open world’ games- even highly linear games like the Half-Life series (let’s be honest: just a series of very twisty, well-designed corridors with puzzles and gunfights) are something you work your way through, learning bit-by-bit what is around each corner and through each door, the first time you play. Learning the shape of each room, discovering the details of each space you encounter, is exploration, and in a sense is a kind of reward or success experience for the brain.
As for multiplayer games, I point back to the earlier question “why do we play (games with rules)?” and decline to elaborate further in this post.
However, engagement with a story can have a totally different kind of value. The argument might be made that ‘earning’ the next piece of a story is a kind of reward system, but that’s not what I mean.
There’s also the simple aspect that you have some thing or things to do while the story unfolds, and can (generally) set the pace, but that too seems like a sort of side-effect.
The recent game that had the strongest emotional impact on me was (no spoilers in this post, don’t worry) Bioshock Infinite. My level of impact on the story was minimal or even zero, but even though my participation and decision making power was limited to a few button prompts with no significant impact, I intuitively felt some level of responsibility for the illusory choices I was asked to make. The story, I think, can stand strong without the added hook of player participation, but both together were able to make a major difference, as far as I’m concerned.
Still, that’s something I don’t accept can be attributed to ‘participation’ alone. No Fallout or Elder Scrolls has had quite the same effect on me personally. The first Dragon Age game, and possibly the first Mass Effect, may have walked that territory a little, but I’d deny that it comes from the depth or variety of player choices.
Well… I suppose that’s all I have to say at the moment, although if anybody would like to discuss Bioshock Infinite, or criticize it, I’d be up for it.
I don’t think it has anything to do with stories AT ALL but I do agree with participation. Most people I know, including me play games because their fun, require skill, visually stimulating and mesmerising, in many aspects like the sound, graphics, difficulty or the manner in which completed parts of it, conquered it in fact. The main thing about everything I just listed is that you get to participate in all of it. I think story is just an added bonus as I rarely think back to the stories of a game and instead remember the soundtrack, how well I played it, or how well used to KO the person sitting next to me.