Download the version for your operating system above, expand the archive, and run the application. You will need Java installed. OpenJDK is OK, that’s why it’s an application rather than an applet (Processing doesn’t work with the OpenJDK browser plugin yet, bizarrely).
The application will ask you for a Twitter username and password. It needs this to connect to the Twitter streaming API and won’t use it for anything bad. You can’t use OAuth for the streaming API yet, so the application really does need a username and password to log in.
If you decide to run the application full screen, you can finish it by pressing the Escape key.
Source code included under the GPL v3.
(Update 2010-06-21 - Thanks to Zeroinfluencer and Laundryman for problem reports on Mac OS X and Windows. I’ve updated the downloads above.)