The idea is to move development into the cloud, by creating a shared environment where code can be accessed through a browser, wherever you happen to be. None of the usual complicated access to corporate networks is required, no heavy-duty development tools, you could maybe even go to work on your mobile (though low resolution and no keyboard means this probably isn't a very good idea).
To access Bespin you'll need a browser with HTML5 and Canvas support, which currently means Firefox 3.5+, Chrome 2.0+ and Safari 4.0+. Create an account at Bespin.mozillalabs.com and you can immediately begin entering code in the upper part of the window.
Then click in the lower command line to enter commands, which help with managing code snippets, debugging, testing, saving files and reloading them later. We don't have the space here for a Bespin command tutorial, but the only one you really need to know is help - enter this in the command line and you'll get a list of available commands, along with clickable links to web resources that will tell you more.