Apple - 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).

I think it should be mentioned that a bulging or swollen battery is a sign of failure and it may be dangerous.

From Wikipedia:

All Li-ion cells expand at high levels of state of charge (SOC) or over-charge, due to slight vaporisation of the electrolyte. (…) This is very noticeable for LiPos, which can visibly inflate due to lack of a hard case to contain their expansion.

LiPo cells are affected by the same problems as other lithium-ion cells. This means that overcharge, over-discharge, over-temperature, short circuit, crush and nail penetration may all result in a catastrophic failure, including the pouch rupturing, the electrolyte leaking, and fire.

I've found a question here on Ask Different referring to similar problems with MacBook batteries (which are basically the same) that explain the possible risks and another one with measures to dispose of a failing battery, after a long rant on Apple.