How do I use Hedera?

I intend, when I have more free time to work on the project, to add an icon to the status bar to indicate that Hedera is enabled and when it is synchronising. For the time being, however, you need only to install the extension for it to do its work.

As it says in the README, in order to see debug output, start Thunderbird with the environment variable HEDERA_DEBUG=1 and open the Error Console. In other words:

me@mine:~$ HEDERA_DEBUG=1 thunderbird

And when Thunderbird starts, go to the Tools menu and select Error Console. Hedera will output quite a substantial amount of information about what it's doing if it's working, or log errors if it's not. You can view the contacts that Hedera has pushed to Couch by opening $HOME/.local/share/desktop-couch/couchdb.html in a web browser and selecting the Contacts database. Note that Hedera does not yet pull the contacts from Couch into Thunderbird, but I'm working on it as I get time.

If you have enabled contact synchronisation between your machine and Ubuntu One (System -> Preferences -> Ubuntu One -> Services tab) then the contacts from your Thunderbird addressbook should end up in the Contacts tab at

  https://one.ubuntu.com/contacts

James Tait suggests this article as an answer to your question:
FAQ #1339: “How do I use Hedera?”.