The port on the back of the Current Cost is an RJ-45, the same as an ethernet port, but it won't simply connect to another ethernet device - you need a smarter cable.

Every six seconds the Current Cost outputs XML along a cable plugged into the RJ45 port as TTL-level serial data at 9600 baud working at 3.3 volts. Through some simple manipulations TTL data can be converted into RS232-compatible data. This means that we need an RJ-45 to serial cable. Once we have that, we can plug it into a normal serial-USB cable and use it anywhere that accepts USB. Hooray! You can order the RJ-45 to serial cable from Current Cost themselves or you can set about building one yourself. An alternative to this is to use an Arduino, which has native USB output so that you can do any data parsing you like before it even gets to your computer. If you also get an ethernet shield then you can connect it to your router and send information straight through to, say, twitter, without requiring a whole extra PC to be switch on!

(The latest CC128 unit outputs data at 57600 bps. So if working with Arduino forget about software serial.)