I suppose the answer is a little different for each of us. The reasons why stories matter to us differ even between each book we like.
For example, A Wrinkle in Time still speaks to me even though the binding is falling apart after having read it about a dozen times (I actually just picked up a new copy in the Spring so that the original one doesn't totally disintegrate). I love Meg Murray. I, like her, cannot seem to find a "Happy Medium". I share in her struggles to control my temper. I often feel like my peers don't understand me.
Another volume I've read just about to death is The Lord of the Rings. And while I skip about half of The Fellowship every time I read the trilogy, I still love it. After I finish it, I feel like I'm ready to conquer the world. Not in a Sauron-esque way, but, to paraphrase the wonderfully wise Samwise Gamgee, to fight for the good in the world that's worth fighting for. I love the books more and more as I grow older, perhaps because it feels like the world's being consumed in darkness and I don't stand a chance against it. But if a hobbit can save Middle Earth, maybe I can make a small dent in my own world.
In my personal opinion, we love stories because they encourage us. Because they inspire us to be better. To change our world. Sometimes, it's easier to get that message from a story than from a real-life experience.
If Meg Murray with her braces and over-reactions can save her brother from IT, if Sam and Frodo can throw the Ring into Mt. Doom even though they're less than four feet tall, maybe we can make a difference.
It's often been said that one needs a willing suspension of disbelief to enjoy a story. Perhaps we can use that same belief in the impossible to defeat the skepticism that so often holds us back in real life.
What are your favorite stories/books and why? Why do you think stories matter?