Answer by Liang-Hai Sie:

What you feel is called Postprandial somnolence.

For ages we believed in the myth that this was because after a meal much blood would be diverted to the gut, so the brain gets less blood, this has been debunked see Debunking a myth: neurohormonal and vagal modulation of sleep centers, not redistribution of blood flow, may account for postprandial somnolence.

 Many people thought that the food composition (high or low carbs causing or not causing a glucose spike after the meal) could influence this after meal sleepiness, this too has been shown not to be true see Meal composition and its effect on postprandial sleepiness.:  no difference between a high carb or high fat meal.

The latest hypothesis is that the production of Orexin, a wakefulness promoting neuropeptide transmitter produced in the lateral hypothalamus, the production of which is inhibited by high blood sugar levels and Leptin (a hormone secreted by the body's fat cells that inhibits hunger), so after a meal there would be less orexin, thus less wakefulness.  See Page on "Metabolic state signalling through central hypocretin/orexin neurons".

Why do I feel sleepy after I have breakfast?