How do polar covalent molecules dissolve in water?

Because water is a polar molecule, any other polar molecule will probably dissolve in it. "Like dissolves like."

Polar molecules lack a symmetrical charge distribution; one half of the molecule will be more positive (referred to as partially positive) and the other more negative (referred to as partially negative). Polar molecules may or may not be made up of polar covalent bonds.

Thus, polar substances dissolve each other because the partially negative part of one polar molecule (such as water) will interact with the partially positive part of another molecule (such as your mystery substance).

This explains, in addition, why non-polar substances like oils and the like have little to attract the polar water molecule.

Polar covalent molecules dissolve in water due to the dipole-dipole interactions between the water molecules and the polar molecules. Water's partial charges attract and surround the polar molecules, breaking their intermolecular forces and allowing them to become uniformly distributed in the water.