Why is my screen bowing out on my iPhone 5?
Nice issue, it happened to me just a couple of weeks ago.
I solved it taking the iPhone back to the Apple Store (3 years old, not covered by warranty). They replaced my iPhone with a refurbished one in 10 minutes for free. I think this is the best solution.
Otherwise you can try to fix it on your own. The issue here is that the battery is collapsing and expanding, pushing your screen away from the case.
Here is a useful illustrated guide for DIY fix. Here is the replacement kit (part + tools).