At the risk of insulting everyone, here is how it works.
There are a pair of steel rails at the bottom of the machine, on these rails run big bearings (sliding bearings). The main bridge sits on these bearings so can run freely the length of the base. If you add a fixed screw to the base, with the nut attatched to the bearings, when the screw turns, the nut moves, taking the whole bridge assembly with it up and down the rails.
Now then, across the top of the bridge there are another two rails, with big bearings, and another screw to move the router along. So you see, by controling the two screws rotation, we can move the router left-right and back-forth - just like a graph, we have an x-y axis.
Of course we need height control for the router, so another set of rails/bearings/screw is added vertically, and it slides on the bridge. This holds the spindle itslef, and is known as the z-axis.
Through the miracles of modern technology, I can take a 3d drawing done in a 3d CAD program, and have the computer break it down into plotted points like a regular graph - x1, y2 etc. The computer now turns the screws to move the spindle to that location. Once it is there, it gets the next location and moves the router to there, and so on.
This is the best photo I have to explain it. You can see the three pairs of rails, and two screws (z-axis screw was not in place at the time). On the end of each screw is a motor, controlled by the computer.
Clear now I hope!