Why are lenses sharper on FF?
It's not that they're sharper, every lens only resolves so many line-pairs per mm and that's it, it's that on FF they appear sharper.
Take an 18MP sensor, it's got 3456 pixels high.
On aps-c (7D etc), that's 3456 pixels in 16mm, pixels are 4.6 microns apart, 216 pixels per mm
On FF (ie 1DX), that's 3456 pixels in 24mm, pixels are 6.9 microns apart, 144 pixels per mm
Maximum (theoretical) resolution should be 1 line-pair in 2 pixels (read here
as to why this is unrealistic in the real world).
Anyway, take a lens that can resolve 72 lppmm, and put it on a 1DX. Take an image of a test chart that has 72lppmm. In a perfect world, that would give you an image that contains 72lppmm.
Take that same lens, and put it on a 7D. The lens can only resolve 72lppmm, but the sensor can see up to 108lppmm.
Blow both of those images up to the same size. The 1DX image contains 72*24 = 1728 line pairs. The 7D image only contains 72*16 = 1152 line pairs (and probably some aliasing and artefacts too). So the same lens on the 1DX looks 'sharper', as it were, it's resolving the maximum that the sensor can handle, whereas on the 7D it's only resolving a lower number.
All of that is theoretical, of course, and a lot of things make those numbers impossible (like Bayer and AA filters, read the above link). Counteracting this, there's also the fact that most lenses get softer near the FF corners, which just doesn't show up on APS-C sensors, so take that into consideration too...