Methanol has a higher boiling point than methane because it has stronger intermolecular forces (IMFs), which are attractions between individual molecules. This makes its molecules harder to separate so it takes more energy, thus the higher boiling point.
Here's how this is related to bond polarity and hydrogen bonding: A bond will be polar when one atom has a greater attraction for electrons than the other does. This causes a charge separation, called a dipole. If there's an opposing dipole in the opposite direction they cancel out and the molecule will be non-polar overall. Methane is non-polar because the carbon-hydrogen bonds have low polarity. If the bonds were polar the molecule would still be non-polar becasue the four dipoles would cancel out due to the tetrahedral bonding geometry.
Methanol looks like methane, with a central carbon, but has one H replaced with an OH creating an unbalanced dipole. Methane is thus a polar molecule. When hydrogen is bonded to one of the three most electronegative elements, F, O or N, the molecule exhibits hydrogen bonding. This is a very strong intermolecular force in which the hydrogen on one molecule is attraction to the oxygen (or F or N) on an adjacent molecule.
Methane is not completely wihout IMFs. All moleules have dispersion forces, which are attractions caused by temporary dipoles due to shifting of electrons. However, the dispersion forces in methane are small compared to the hydrogen bonds in methanol.