I work for Anyfi Networks.
The VPN tunnel carries raw Wi-Fi frames from the visited Wi-Fi router to the home Wi-Fi router. The visited router only acts as an antenna - no traffic is ever terminated there. You can think of the VPN tunnel as a very long antenna cable connecting new antennas to the home router on demand. The visiting user can never access your LAN, your encrypted Wi-Fi or use your IP address on the web.
As a visiting user you will get a public IP address from your home so no one else will get the blame for anything you do on the Internet. The WPA encryption goes all the way from your client device to your home so the owner of the visited router cannot eavesdrop on your data. From the client device point of view, you are at home.
The PR is primarily geared towards ISPs, but the technology as such is available to anyone. If you are comfortable with OpenWrt you can download firmware from
http://anyfi.net/getit/firmware for popular consumer Wi-Fi routers and try it out yourself. There is also an FAQ
http://anyfi.net/faq.
BTW, "residential gateway" is ISP lingo for the modem/router installed in the subscriber's home.