So I don't need to buy a desktop version of Office to use the Web Apps?

Correct. However, you will be able to purchase the Office Web Apps through Microsoft Online Services. It's unclear why you'd want to though.

Are the Office Web Apps as good as their desktop counterparts?

No. They're designed to complement the desktop apps. You might, for instance, use the Web Apps to save files on Microsoft's online storage site. Or you could access the Web versions while working remotely from a kiosk, netbook, or other Internet-connected PC.

What's the deal with the free Office Starter Edition?