This Live Stream Plugin is really just an iframe. The content of the iframe is the live stream content.
So the Live Stream button looks like it's on your own site, but it's really being pulled from Facebook's site.
Notice that the Application ID (the unique id associated with every Facebook App) is being passed in the query string to the Facebook server along with the request for the iframe.
Here is the code used on this page:
<iframe src="http://www.facebook.com/plugins/livefeed.php?app_id=75460339721&width=400&height=500" scrolling="no" frameborder="0" allowTransparency="true" style="border:none; overflow:hidden; width:400px; height:500px"></iframe>