Multiple patents exist around file (specifically picture) sharing from mobile devices, yet mobile picture sharing apps are springing up everywhere and receiving a considerable amount of funding from VCs and angels.  Can anyone explain why?

You need to distinguish the concept of a "broad idea" (say, file sharing) with a patent, which describes a very specific way to implement that idea. The fact that someone invented a mouse trap or an electronic mouse never stopped people from inventing new mousetraps or new electronic mice.

Assuming the premise of this question is accurate, I think the answer would simply be that investors have found that the potential return on investment for a picture-sharing app far outweighs the risk of potentially incurring patent infringement liability. I also think it would be nearly impossible to avoid the potential for liability altogether in this type of technology, thus favoring the approach of dealing with infringement claims as they come along (if they even do). There are areas where I imagine pre-investment infringement diligence would be more important. These might include pharma, biotechnology and medical devices, where it's relatively easier to narrow down what patents exist, who owns them, and who is likely to assert an infringement claim. For mobile apps, though, it's probably just not worth the effort. One could spend 100 hours trying to identify all potential infringement risks for a mobile app and still come woefully short of getting any kind of certainty.