The principle is that while they may seem to move in any direction from the point of view of the observer, they radiate from a common point of origin.

If you had a piece of graph paper and an accurate plot of their various trajectories, and you extended the lines backwards, you would find a point of convergence [edit: divergence, actually]. In this case - somewhere the constellation Leo, which is the reason they are called Leonids.

(One of my early Astronomy One lab assignments was to do exactly that, though I think it may have been the Perseids [August], not the Leonids.) I'm starting to sound like you, Bill! :-)