Learn English – Difference between “larder” and “pantry”

Etymonline has this to say on quarters:

"military dwelling place," 1590s, from quarter (n.) in sense of "portion of a town." The military sense is in quartermaster (mid-15c.) and might be behind the phrase give (no) quarter (1610s), on the notion of "to provide a prisoner with shelter." The verb quarter "to put up soldiers" is recorded from 1590s.

Quartermaster appears to be from something unrelated to fourths:

mid-15c., from Fr. quartier-maître or Du. kwartier-meester; originally a ship’s officer whose duties included stowing of the hold; later (c.1600) an officer in charge of quarters and rations for troops.

Although I am weak at French, it appears that quartier means area or district. Note that the dates for quartermaster are before quarter: mid-15c versus 1590s. This implies that quartermaster transferred first and then people began using quarter due to the use of quartermaster.

Transferring the meaning of quarters into a non-military dwelling place seems reasonable and, therefore, the short answer to your question is that quarters appears to have come from France by way of quartermaster.

Edit: My hunch on why quarter came from quartermaster (as opposed to Etymonline just telling us it did) is the inclusion of master in the term led people to assume that they were masters of something; it was logical to call that something a quarter. The idea of quartering up soldiers would mean to provide for them; this would be the quartermaster's responsibility. Lodging was most likely included in that and the perception of "master of lodging" makes sense.

Leverage means to use something in such a way that a small amount has a big result, much as a lever lets you move a larger load than you would normally be able to. In terms of the statement, Zend can be made more effective with plug ins. That is to say, the plug in means that your web app gets extra functionality from Zend+plug ins than it would from Zend alone.

It is frequently used in finance to mean that a small amount of money (like a house downpayment) can be used to much greater financial effect, such as buying a house.

Utilize means something completely different. Utilize is often used to mean "use" but it does have a subtly different meaning, specifically, utilize means use effectively. So you can load the Zend framework, but still write all your own code in javascript. But if you utilize it, you use the Zend functions effectively to get the results you want.

Utilize is certainly an widely abused word, but that doesn't mean it can't be used correctly, or should I say utilized correctly.

According to dictionary.com (my emphasis):

utilize
to put to use; turn to *profitable* account: to utilize a stream to power a mill. 

You can make the replacement, but it would change the meaning.