That one sentence is: When nuclei get bigger, they get bigger, but when atoms get bigger, they get smaller.
What is meant is this: As you move up the periodic table, you get more protons and more electrons. This means the attraction between the protons and electrons grows, and the electrons are pulled closer to the nucleus. (There are exceptions—we are talking trends.) Thus the paradoxical statement: when atoms get bigger (more charge) they get smaller (electrons pulled closer.)
Nuclei, on the other hand just get bigger when they get bigger.
This means the nuclear force that binds nuclei is fundamentally different than the electromagnetic force that binds atoms. Long range forces like the attraction between charges (which is of infinite range--though only weakly, a proton on one side of the solar system will feel the force of an electron on the other) cause the behavior we see in atoms. Short range forces, that are very strong only at short distances and zero at larger distances, cause the behavior we see in nuclei (bigger means bigger.) Why? Because the nuclear force has a short range, something like the size of a proton, neutrons and protons in a nucleus see only their nearest neighbors. Adding more neutrons or protons doesn't increase the attraction--because they don't even "see" most of their colleagues.
Thus this simple observation teaches us that the nuclear force is short ranged, which is the beginning point of all nuclear physics.
UPDATE: I've been a bloggin' too long and I'm getting too long in the tooth--I forgot that I had already blogged about this--only much better.