I followed the article "Tony Fisher's Cuboid 3x3x3 Transformations" posted in this site up to the point where the incisions are shown. I must be dense. Why are they needed?

Hi,

The Incisions are needed because it is now part off the internal mechanisme and the outside cubies will overlap the internal cube part. So the incisions are needed else you could not turn the cube after putting the outside cubies in place. Normally the inside will be round! In this article it is simplyfied by just make incisions, if you want to do it right, all incisions will be part of a circle.

Ton

Or to put it another way, unless you make the incisions, turning a face 45 degrees will press your outer cubies into the edge of the inner cubies. Therefore, your outer cubies will pop off or break.