I'd guess he was talking about SN2 reactions with neutral nucleophiles. Anionic nucleophiles are more reactive in polar, aprotic solvents because the anion is poorly solvated (less stabilized by solvent). e.g. Dimethylsulfoxide (CH3SOCH3; DMSO) solvates cations well because the negative oxygen is unhindered. However, it solvates anions poorly because the positively charged sulfur is crowded.
There are neutral nucleophiles that can participate in SN2 reactions (SR2, NR3, PR3), and I believe they react faster in polar, protic solvents. There are undoubtedly still differences in solvation of the nucleophile between protic and aprotic. However, I believe the bigger issue here is that the product is now a salt, which is stabilized by solvents that can solvate both the anion and cation (protic).
In a nutshell, for anions, the starting material is made more reactive/less stable by polar, aprotic solvents. For neutral compounds, the product is made more stable by polar, protic solvents.