TL;DR: This is a proxy server that routes HTTP requests from a unique subdomain (e.g. http://m7rp3u2ntr9t3xr.browserver.org) to a browser that's visiting browserver.org. The JavaScript in the browser then handles the HTTP response. This is possible thanks to a persistent connection between the browser and browserver.org, via some "websocket-like" protocol.

The title "node.js HTTP server, in your browser" is because the HTTP handler in the client (browser) has a Node-like API, see:

https://github.com/jed/browserver-client#example