Nylons are a class of polymers formed from di-acid and di-amine monomers. The amine groups bond with the acid groups to form amide groups, end to end. The resulting amide groups have a positive charge at the amine hydrogen, and negative charge at the acid double-bond oxygen. The result is a polymer with periodic polar moments across the chain. The polar moments align between neighboring chains, creating a strong van der Waals force between them, in the form of hydrogen bonding. This is part of why nylon fibers are so strong: they resist chain sliding due to the strong inter-chain bonding.
Water is also a polar molecule and also exhibits hydrogen bonding, giving it some interesting properties compared with similar molecules. Because of its polar nature, water readily infiltrates nylon materials and is attracted to the polar amide groups of nylons. Water creates space between chains, increasing the ability for chains to move around one another more easily, improving ductility. As Chris Johns noted, it acts as a plasticizer.
It is worth noting that nylon and other amide polymers are somewhat special in that they are hydrophilic and oleophobic, unlike other common engineering polymers.