A fair approximation of something means a moderately good imitation of that thing. They are doing the actions of studying, spreading textbooks and notebooks around and generally creating the appearance of students hard at work.
To break out a thing means to bring it out of storage or provide it to others. It's typically used for food and drink or for an object that has a particular role or that is used periodically for a specific function. In this case, the books have been gotten out for the purpose of studying.
In short, the students have gotten their study materials out, spread them around, and to any outside observer, are studying hard. However, they have not made any progress and only appear to be working.
John spent several hours trying to copy the cartoon's artwork. In the end he produced a fair approximation of their work, but it was far from perfect.
Stephanie's cake was a fair approximation of the one shown in the magazine, particularly considering her lack of skill as a baker.
It's New Year's Day! Break out the champagne!
Break out the broom and dustpan, here comes that kid again. He breaks something every time he comes in the store!