A proxy server can reroute traffic from an online game in the same way as any other traffic, including web pages, by accepting data on one end, tben passing it on to the recipient and vice versa. It is a very simple process. All of your networking equipment constantly does the same thing.
A lot of web proxies use a slightly different setup that does not require the client to make any changes to his configuration in order to have traffic not ultimately intended for the proxy server to pass through it nonetheless. Instead, technically the proxy server itself is the recipient, the location of the website the user wants to see is merely content sent to the server. The proxy processes this request by browsing to this location, then responds with the downloaded contents. Services such as Hide My Ass! are examples of this. Naturally, this type of proxy only works for websites.
The latter setup is obviously much more common. Relaying any traffic without regard for its content, usually in the form of a VPN, is a service not many would provide free of charge. And indeed, most VPN providers only allow paying customers to access the internet through their network. The players of your game might be using such a service to circumvent their IP-bans, but it is unlikely the VPN server can provide them with multiple external IP-addresses; they'd either be using their own, or the one from the VPN server.
It is very well possible to get a new IP every time using this method and there are plenty of other ways to communicate with an online game via someone else's network, but the equally likely and far simpler explanation is that their ISP hands them a dynamic IP and they're not using a proxy at all.
Is there a definitive list of IP addresses that could be added to the game's blacklist? – Hand-E-Food – 2012-08-16T06:14:10.583
Definitive - probably not. But there are blacklists containing known tor exit nodes. – AnonymousLurker – 2012-08-16T07:46:20.247