To expand on the recap:
To expand on why 'splitters' don't work in normal usage:
The cable used for Ethernet, commonly called 'Cat-5' (sometimes Cat-5e or Cat-6 when they're made to higher specs,) has eight wires in it.
100Base-TX Ethernet only uses four wires. They are 'pins' 1 and 2 as a pair, and pins 3 and 6 as a pair. These are the four pins that every normal Ethernet device on the planet try to use. The other wires are unused, and no normal Ethernet device (like the AirPort Extreme Base Station) will try to use them. Indeed, these pins are usually not connected to anything inside devices.
If you already have a Cat-5 cable running some long distance, and need to get a second Ethernet connection on each end without running a new cable, you can use the SPARE four wires in the cable to run a second Ethernet connection.
This is not considered 'normal', and technically violates the 100Base-TX Ethernet standard. However, this is what the 'splitter' is used for. It assumes that the other end of the cable has been wired so that the spare four wires connect to another splitter or similar. So BOTH ends of the wire need to accept two Ethernet connections. (Which the AirPort Base Station does not.) This splitter just uses the 'standard' set of four wires for one plug, and the 'extra' set of four wires for a second plug. If the device at the other end doesn't know to look for something on this extra set of wires, then it won't work.
A common use would be in businesses where they have their Cat-5 wires all going into a central server room, which then runs through a 'patch panel' to map all the cables more intelligently than just a wad of wires. If the extra four wires were 'patched' into a separate jack in this server room, then a splitter would work, as these extra four wires now connect to something on the other end. It WON'T work with any standard Ethernet equipment, though.
Finally, 'Gigabit' Ethernet, or 1000Base-T, uses all 8 wires, and using such a 'splitter' would break Gigabit compatibility, forcing any computers connected this way to drop to 100 Mbit speeds. (Or worse. I have seen techniques like this used that even breaks 100 Mbit speeds, forcing computers to drop to 10 Mbit.)