Why can't two computers just be linked up, with some sort of virtual server set up between the two, to play multiplayer games?
They can, it's called a LAN (local area network). Most computers these days are connected to one in some way or another by usage of a router or hub. Many multiplayer games support LAN play, and those that don't can generally be tricked into allowing it by usage of Hamachi -- which is basically what you asked for, a virtual private network. It's not limited to LAN usage though, it works both ways, it can trick games with LAN-only multiplayer into thinking people from all over the internet actually belong to your local network.

You've probably used ethernet cables before if you've ever used a router. These types of ethernet cable are actually one of various cable configurations called 'patch cables' which basically require a middle-man such as a hub or router. But there's another configuration of the same type of cable (with one or two wire pairs swapped) called a "crossover cable" that allows you to connect two computers directly together using existing network cards.