By "everything is moving at c," a 4D spacetime is referred to rather than a 3D traditional space. Basically, if we are not moving in the first three dimensions relative to an outside observer, then we are moving completely in the 4th dimension. As we move closer to a total of c in the first 3 dimensions, though, less of our "4D-velocity" is directed in the "t" direction and therefore we appear to experience less time to an outside observer.
Basically, this is just a simple way to try to explain the effects of relativity; that is, if we move at a velocity closer to c in space, we move at a slower "velocity" through time.