I am by no means a biochemist/physiologist, but I will give this my best scientific go.
The main reason people get sleepy after eating large meals, "get the 'itis" as many call it, simply has to do with homeostasis. The average person's diet usually consists mostly of carbohydrates. Carbs, especially when eaten in large quantities, causes a large spike in our blood sugar. In order for your body to bring the sugar levels back down to normal, the body releases insulin.
This stimulates movement of tryptophan (an amino acid in the brain), which then leads to the increased production of serotonin, a neurotransmitter responsible for sleep. And this increased level of serotonin triggers sleepiness after a meal. - See more at: http://blog.zocdoc.com/get-sleepy-meal-weird-science-daily-life/#sthash.fseXW9aA.dpuf
Dangerously enough, constant spiking and falling of the blood sugar levels in the human body can lead to T2 Diabetes (most associated with processed sugars).
Tryphtophan is the same amino that is found abundantly in turkey may be why some people take snoozes after large Thanksgiving day meals!
Now, this brings us to the coffee portion. Coffee has caffeine which is a CNS stimulant. It increases blood pressure, metabolism, causes natriuresis (increase in saline secretion), but most obviously, may cause jitteriness, mild anxiety and alertness.
This is because when caffeine is consumed, it antagonizes adenosine receptors; in other words, caffeine prevents adenosine from activating the receptor by blocking the location on the receptor where adenosine binds to it. As a result, caffeine temporarily prevents or relieves drowsiness, and thus maintains or restores alertness.
It is not the blood in your body that affects sleepiness/alertness, rather the neurotransmitters and their chemical synapses
TL;DR: After eating, your body is naturally producing neurotransmitters to bring your body to a state of homeostasis. When you ingest coffee, which contains caffeine, it counteracts this and keeps you awake and alert.