Pharmacokinetics: why do certain drugs follow zero-order kinetics?
Basically any compound in concentrations sufficient to saturate its metabolization machinery will show zero-order pharmacokinetics. This reflects the fact that metabolization is taking place at full speed while facing a comparatively enormous amount of substrate (just as @WYSIWYG pointed out). This effect can also be shown in vitro with pure substrate/enzyme solutions.

But the above is a rather simplistic model, valid only in specific situations.

Most of the time in humans, zero-order kinetics stem from the fact that liver enzymes are saturated. However, for most drugs the redistribution phase is much more important than metabolization/elimination in defining the duration of effect. A good example of that would be sodium thiopental. As a single iv bolus dose, it will induce loss of consciousness for a short time due to redistribution. However, continued infusion will allow blood and tissue concentrations to equilibrate, and then the metabolization/elimination rate will become much more prominent in defining the duration of effect. And as sodium thiopental happens to have zero-order metabolization kinetics in continued infusion, it will take you a loooooooooong time to wake up! This is why in human pharmacology, we use context sensitive half-life to better represent drug kinetics.