How do polar solvents increase nucleophilicity?
Polar solvents could be protic (hydrogen bonding) or aprotic (no hydrogen bonding).
Polar solvents either decrease the nucleophilicity or not affect it, but they generally do not increase nucleophilicity.
Polar protic solvents solvate the nucleophile through hydrogen bonding and shield it from reaction, which will result in weakening its nucleophilicity.
Polar aprotic solvents do not solvate the nucleophile and therefore, the nucleophile it is said to be naked , which will not shield it from the reaction, and therefore, not affecting its nucleophilicity.
Images source: Organic Chemistry-Janice Gorzynski Smith 3rd Ed