To summarize the approach in the PDF, they reverse-engineered the wire format used to update the client. The extension encrypts small sections of the document (as small as individual characters) with block ciphers. The ciphertext is then sent to Google over the reverse-engineered wire format.

Since they aren't sending actual text to Google, they don't have access to server-provided features like spellchecking or exporting, and they didn't try to handle simultaneous collaboration. However, they assert that if you share your extension's settings for a particular document, your collaborators can all get updates, they just can't type when you type. They also found they hit Google's document size limit pretty quickly, since encrypting individual characters makes the document 16x larger.

In exchange, they claim this gives them enough security to update documents over an insecure channel (they cite China blocking https access to Docs as a compelling example), and Google only stores an encrypted form of the document, so the contents are unrecoverable by Google.