Why do academics work only nine months of the year?
The following applies to a previous version of the question: "Why do academics get paid to work only nine months of the year?".  The question of whether academics do work work in the summer, compared to whether they are paid to work over the summer, is more difficult.  Most faculty in the U.S. have 9-month contracts, but in my experience most faculty nevertheless work on their research and teaching during the summer. 



In the United States, it is very common - nearly universal, in fact - for a standard academic contract to span 9 or 10 months. There are several reasons for this:


It allows the university to pay less. The standard rule of thumb is that a 9 month contract pays 9/12 of what the corresponding annual contract would pay. Many (most?) faculty have their 9 months of pay split up into 12 months of paychecks -- but this is an accounting fiction, not a sign that they are paid for 12 months of work. If few students are around over the summer, there is less reason for universities to pay salary to their faculty for that time - having a break in the contract balances the university income and expenses.
It allows faculty to be paid more. This is the flip side of paying 9/12 of an annual salary: the faculty member can, in principle, take another job during the summer.


Many grants, for example, are able to pay "summer salary", which is really just extra income. But these grants do not allow the faculty member to take salary from the university at the same time they take salary from the grant. The "9 month contract" resolves this: the faculty member can state they are actually only paid for 9 months of university work, so the grant can pay 2 or 3 months of extra salary during the summer.
Similarly, some universities pay extra to faculty who teach during the summer - that would not be possible if the summer was treated as part of the basic contract.
A smaller number of faculty use the summer for consulting work, or other jobs genuinely different from their university position.

Faculty like to travel over the summer. During the academic year, it is hard to leave for personal reasons, because of class. By claiming that the summer is "off contract", academics can travel however they like during that period of time, without having to justify their time.  So having the contract pause during the summer makes the job more attractive to job candidates, allowing the university to attract some people who might otherwise take higher-paying jobs elsewhere.


The reason I just cited are all purely economic - they can be justified solely in terms of saving money for the school, increasing pay for the faculty member, or attracting better candidates. There is also "tradition" as a reason for the summer break, as is also the case at lower levels of education. 

However, the "9 month contract" is fictitious in a few ways:


Most faculty still have access to their office, library, email, and other university resources even when they are technically not "on contract" over the summer. 
Research faculty usually continue to work on research during the summer, and travel to conferences (which may even be reimbursed by the university, even though the faculty member is technically not on contract!). This often happens even when the faculty don't have grants or other summer funding. 
Many faculty use some of their summer time to prepare for their classes, even though they are not being paid to teach over the summer.
There are other activities during the summer: committee meetings, advising, etc. Some faculty participate in these, for various reasons, even when they are off contract.