I think the quoted text is aiming for the following (not that it was good in communicating this):
Let’s consider the torus and the phase-space volume enclosed by it. This volume must be invariant under time evolution:
No trajectories can move out of this volume. Trajectories on the surface stay there (due to it being a solution). Trajectories from the inside of the torus stay inside (because they cannot intersect the surface). The latter point is why this argument does not translate to limit cycles.
Every point inside the torus has an appropriate predecessor (just invert time and you’ll find it), i.e., there are no holes appearing under time evolution.
However, due to the ubiquitously negative divergence, this volume must shrink. This is a contradiction.
Now, what about phase-space volumes “shrinking to the surface” (for an attracting surface of the torus)?
If you considered a sufficiently small neighbourhood of the surface, this would indeed shrink over time.
However, this does not apply to the entire basin of attraction.
As already mentioned above, every trajectory in the basin of attraction can be traced back backwards arbitrarily to another point in the basin of attraction.
Thus the basin of attraction does not change its volume under time evolution.
Another way of looking at this is:
If the surface of the torus is attracting points from its interior, it is a sink of phase-space flow.
Thus there must also be a source of phase-space flow within the torus (e.g., an unstable limit cycle).
This source must have a positive divergence (because that’s exactly what divergence measures).
Note that the ubiquitously negative divergence of the Lorenz system is a special property not found in many dynamical systems.
Most dissipative dynamical systems are only dissipative on average or for specific regions of interest, but have source of phase-space flow somewhere.
Finally note that you can make an analogous argument for a two-dimensional system (with ubiquitously negative divergence) and a limit cycle, if it helps visualising.