It looks like it has damaged the finish of the metal and glass.
What have you tried so far?
Looks like damage to the anodization coating. Unfortunately, there isn't much you can do about it. Some members just touch up with a marker but this was with Black and Slate iPhone 5 and hence easier to color match.
Definitely one of the situations where cases do more harm than good and I've seen quite a few of them in that condition on ebay.
I hope this isn't the case...I don't get the logic in people buying expensive phones with cheap $5 cases for them.
It doesn't necessarily have to do with the case itself. If stuff gets in the case it's liable to do some damage to the anodization coating, no matter how expensive the case is.
On the black glass portion only I've used a microfiber cloth with a dab of water, I didn't want to try anything else before asking.