On the client (the Mac) they have a IOKit ethernet driver that shows up in Network Settings. This becomes your default route (after some fake DHCP magic) and then captures packets and relays them to their application that runs in userland.
Their userland application contains a lightweight TCP/IP stack that interprets those packets. (For example http://www.sics.se/~adam/lwip/)
The lightweight TCP/IP stack contains unimplemented callbacks for opening/closing sending/receiving data. In this case those callbacks are implemented as a higher level network protocol.
That 'wrapped' TCP/IP protocol is then relayed to the iPhone via the usbmux protocol. This allows you to map a TCP/IP service on the device to a local port on the Mac. (I happen to have written a nice usbmux proxy three years ago)
So now the wrapped TCP/IP is arriving at the iPhone, in the form of higher level 'network stack commands'. The iPhone app now simply has to interpret these commands and translate them into real socket open/close read/write commands.
At this point packets are flowing through a fake Ethernet device driver to a secondary TCP/IP stack, over a USB cable to an iOS app which will do actual socket calls and talk to the real network.
Soooo, my guess is that for this new app they use basically the same setup except that the iOS app is gone and replaced with a simple web page with web sockets that point to both that fake TCP/IP stack on the mac and to a hosted proxy service on their end. Relaying data back and forth.
Crazy hack. Amazing it works :-)