How does hydrogen bonding affect the boiling points of chemical compounds? My main question is why is the boiling point of methanol so much different from that of water? 

I understand that both compound are able to develop hydrogen bonding, and obviously water can develop one more hydrogen bond than methanol. 

But on the other hand:


  Due to presence of an electron-releasing alkyl group, the oxygen atom of an alcohol molecule is less-electron withdrawing on the remaining hydrogen atom. Therefore the electron deficiency of the hydrogen atom is less than that of a water molecule resulting in the formation of a weaker hydrogen bond. 


What does this actually mean? 

Both hydrogen, in methanol and water, are connected to oxygen, which has the same electronegativity and takes electrons off the hydrogen making it have a partial positive charge. 

Why does a "less electron withdrawing oxygen atom" lead to a weaker hydrogen bond? 

I thought that a less electron withdrawing oxygen atom would mean that the H is more negatively charged, meaning it would be stronger for hydrogen bonding, since F, O, N want more electrons as they are more electronegative?
A hydrogen bond is a particularly strong dipole:dipole interaction with some covalent character. If we ignore the covalent character for a moment, we can say the higher the partial positive charge on the hydrogen, the stronger the hydrogen bond (keeping the O-H bond distance constant). If you relate "electron deficient" to positive partial charge, it might help you to make sense of the textbook answer.

You write:


  Both hydrogen, in methanol and water, are connected to oxygen, which has the same electronegativity and takes electrons off the hydrogen making it have a partial positive charge.


Yes, that is the primary reason compounds like water, methanol and - say - hydrogen peroxide and acetic acid are hydrogen bond donors. But the chemistry beyond the OH group also plays a role. To illustrate, just compare the acidity of the proton connected to the oxygen in these four cases. They are very different. In a similar manner, the hydrogen bonding will be different (not as different, though).