They're backed by Sequoia so definitely must have a monetization plan. Sequoia wouldn't drop money on them without looking for a return on their investment. Everything seems to be free on their site at the moment and I'm really curious to learn how they could potentially monetize.
Piazza is currently focusing on user experience and driving growth. They haven't begun to monetize yet.
If they're able to execute and significantly grow the userbase, figuring out how to monetize won't be an issue. The target audience (students) is a well sought after market. Some ways they could possibly generate revenue:
Piazza currently has something called "Piazza Careers" in Beta.
As a student, you can upload your resume and build your profile by asking good questions in your classroom's Piazza. Piazza can essentially sell your Piazza profile to employers who are looking to hire talented potential employees early on when they're still in school, before other companies can get access to them.
Employers will benefit by gaining access to talented students, students will benefit by gaining job opportunities, and Piazza of course will monetize. Everyone wins in this business model.