Significance & applications of polarizability for nucleophiles
The examples the site you mentioned gave of uncharged nucleophiles were: the hydrides of Group V and VI. They say that the bigger atoms can have a more effective overlap of atomic orbitals and thus, the bigger the uncharged nucleophile, the stronger it is. 

I would like to dispute this reasoning and answer your question regarding this trend. 

How we should see a nucleophile should always primarily be in terms of electron donation because a nucleophile donates electron density to electrophiles to form bonds. This is a very important consideration.

For bigger atoms, the valence electron pair, which is being donated to the electrophile, is further away from the nucleus and is more shielded from the nuclear charge. Thus, it is easier for donation to take place. This should be the reason why uncharged nucleophiles with bigger atoms are more nucleophilic; that is, they are more capable of donating the electron pair of theirs. My explanation of the trend has nothing to do with polarisability.

The reasoning stated is incorrect because although bond formation is easier for bigger atoms, it does not mean that the overlap of atomic orbitals is more effective. In fact, it is likely less effective because the atomic orbitals used are more diffused (Recall that as the principle quantum number n increases, the orbitals become more diffuse in terms of electron density). Thus, their reasoning is nonsensical.

Hope I have clarified everything you were not sure of.