Methanol Structure
The carbon atom in methanol is bonded to three hydrogen and one oxygen atoms; the oxygen atom is in turn bonded to the fourth hydrogen atom. The carbon atom and its three hydrogens form a pyramid with the carbon at its apex; the oxygen atom and its attached hydrogen are positioned like a flagpole and flag sticking up from the top of the pyramid. You can also imagine methanol as a tetrahedron with the carbon atom at the center, three of the hydrogen atoms and the oxygen atom at the corners, and another hydrogen attached to the oxygen atom.
Electronegativity
Electrons in bonds between different elements are often shared unequally. You can get a sense of how they will be distributed by comparing the electronegativity of the two elements, which is a relative measure of their ability to attract and hold onto electrons. As you head up and to the right across the periodic table toward fluorine, elements become increasingly electronegative. Oxygen is much more electronegative than either carbon or hydrogen, so it's more of an electron hog and tends to keep the shared electrons closer to itself.
Polarity
Methane is completely symmetrical. The carbon atom forms the center of a tetrahedron with a hydrogen atom at each of the four corners. Carbon and hydrogen have very similar electronegativities, so they tend to share electrons almost equally. This makes methane a nonpolar compound -- a compound with no charge separation. Methanol, by contrast, is highly polar. The hydrogen attached to the oxygen atom is electron-poor and thus somewhat positively charged while the oxygen is electron-rich and thus somewhat negatively charged.
Hydrogen Bonding
This disparity creates links called hydrogen bonds between molecules of methanol. The hydrogen atom on one molecule of methanol can interact with the oxygen atom on another. These interactions are nowhere near as strong as the bonds within methanol, but they help to hold the different molecules together. It takes energy to break the hydrogen bonds, so you need much more heat to boil methanol than you need to boil methane. Methane, by contrast, is nonpolar and has no hydrogen bonding, so the attractive forces between molecules of methane are much weaker, and it boils at a lower temperature.