The big rip is not generally considered a plausible scenario any more,since we now have a better understanding of cosmological expansion.
However I think the general idea is that gravity would at first hold more massive objects together,
While galaxies are big they are not very dense, the gravity holding the individual stars of as galaxy together is tenuous.
By contrast the gravity holding the atoms together in a star is stronger, but very local to the star.
The final outcome in that scenario is that dark energy having defeated gravity eventually goes on to defeat even the strong nuclear force holding atomic nuclei together.