As you'd expect, science has some answers. It turns out that there are two big factors that make you fancy a snooze soon after the pumpkin pie is polished off.
First, when the food starts to arrive in your belly, the activity of the parasympathetic nervous system increases and the activity of the sympathetic nervous system decreases. Huh, what does that mean? Well, the sympathetic nervous system provides our fight-or-flight response. The parasympathetic system gets your organs ready for digestion. Basically, your body wants you to stay still in order to sort out the contents of your stomach. That's why you choose not to move too much after your meal.
Next, when you start digesting your food, you get a big rush of glucose into your blood stream. If you don't have diabetes, your body creates insulin to help the body's cells absorb the glucose. The insulin works by affecting the uptake of a bunch of amino acids in the body — stick with me here! — except for one called tryptophan. So the concentration of tryptophan increases relative to other amino acids. Turns out that in the brain, tryptophan is converted to serotonin, which is itself converted to melatonin — both of which result in sleepiness.
Voila! Your nervous system and your brain both want you to sleep. So I say go with it.
Incidentally, there's a myth that turkey contains a lot of tryptophan. That's kinda rubbish, though, as turkey doesn't contain any more than chicken, beef, or plenty of other meats. So don't blame the turkey too much. Blame your gluttony instead. More here.