I understand that alcohol and phenytoin are two examples of drugs that follow zero-order kinetics. Why do these two particular drugs follow zero-order kinetics as opposed to first-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.
Alcohol is a drug that is cleared by an enzyme. This interaction is what follows zero-order behavior. However, initially alcohol follows first order behavior because the ability of the enzyme to clear the alcohol exceeds the concentration of alcohol. That means clearance is constant and does not depend on the alcohol concentration.
Now what makes alcohol zero order is that the desired effect of alcohol (the therapeutic effect) occurs when the dose is greater than that enzymes ability to clear the alcohol. Once the alcohol concentration is greater than a certain point the body begins to struggle to clear the alcohol (roughly near the Km of the clearance enzyme) because of a physical limitation on the quantity or quality of the enzyme. So now when you increase the concentration of alcohol, the rate of clearance is not able to account for that increase proportionally because it is saturated. Therefore the alcohol you consume after a certain point (Km of the clearance enzyme) has a greater toxic effect. One drink too many is a very real thing.