I haven’t looked too closely at the algorithm you’re referring to, but my understanding it this:
You take two meshes and project the vertices out onto a sphere. You can then look at how the vertices land on the sphere to determine correspondences between vertices on the source and target mesh. Then you just animate the vertices moving from their initial to final positions, and voila! A morph.
It’s a nifty idea and probably pretty simple to implement. Just keep in mind that it has a lot of shortcomings too. The biggest being that objects which don’t project well onto a sphere (objects with complex topology) are going to give wacky results. Using this algorithm to morph between a sphere and a torus, for example, would probably not work well.
Oh, and to finally answer your question, a unit sphere is a sphere of radius 1. For the sake of the algorithm, that’s not really that important.
Once again, let me emphasize that I’m not 100% familiar with the algorithm. This is just my recollection of how it works. I’m sure there are details and complexities I’ve omitted.