Why is the EM5 so much more popular than the GH3?

Yes, size - that thing is huge. I see very few reasons to choose GH3 vs Nikon D7000, D5200 or Pentax K5, especially if video is not important.

By choosing GH3 vs top APS-C DSLR you lose: PDAF and focus tracking, OVF, larger sensor with better DOF control and IQ. Lose money too, as GH3 is more expensive than most APS-C DSLRs

You gain only small weight and size saving in lenses.

By choosing E-M5 you lose the same things but you make a jump into much smaller form factor. The small form factor allows to carry camera more often, and bring camera to places where DSLR wouldn't be appropriate or convenient. So, it's that and IBIS, which gives stabilized viewfinder.

GH3, on the other hand, is just an expensive DSLR without DSLR guts and features.