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The absence of CO2CO_2 in your blood increases the pH level from its normal balance between 7.35 and 7.45 to 7.5 or higher, making it alkalotic. Since we’re doing this by breathing, this process is called respiratory alkalosis. This alkalosis triggers a Rube Goldberg Machine of biochemical reactions that lead to tetany and “tingling”.
Rube Goldberg Machine
First, the alkaline pH affects a protein called albumin. Albumin is the pick-up truck of your blood, carrying all kinds of hormones, fatty acids, and other compounds through your blood. One of those is calcium. During alkalosis however, albumin tends to bind more strongly to the calcium. The total calcium contents of your blood doesn’t change, but because albumin hangs on to the calcium it carries, the amount of free-floating, ionized calcium (Ca2+) in your blood decreases. The fancy word for this is alkalosis-induced hypocalcemia. Calcium plays an important role in regulating how active our neurons are. Calcium normally stabilizes the so-called voltage-gated sodium channel, making sure not too many sodium ions get into the club. When albumin snags up all the calcium however, neurons are much more easily excitable. This affects especially sensory neurons and motor neurons. Sensory neurons are neurons that react to an outside stimulus (like neurons in your skin that react to touch) by firing and sending this signal to the brain. When these neurons become more excitable, you may have a sensory sensation without being actually touched. This is generally what produces the sensation of tingling during CCB. Motor neurons on the other hand are neurons that are connected to our muscles. When they receive the signal from the brain to fire, they release a neurotransmitter (acetylcholine) onto the muscle, which causes the muscle fiber to contract. If motor neurons are more excitable, your muscles can involuntarily contract, which is called tetany. This effect is mostly felt in the hands and around the mouth (the perioral region if you want to be fancy). The reason why these regions are affected first isn’t entirely known, but it is likely a combination of smaller muscles and higher density of nerves in these areas.

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Effects on blood vessels and oxygenation

Apart from the effect decreased CO2 concentration has on your peripheral nervous system, CO2 is also what we call a vasodilator, which means it expands blood vessels. The exact mechanism is not entirely clear, but the main theory is that CO2 relaxes the smooth muscle cells that line the blood vessels, allowing the vessels to expand. This effect seems particularly strong in cerebral blood vessels (ie, in your head). In the absence of CO2, blood vessels constrict, limiting blood flow. Additionally, the increase of pH in your blood inhibits the release of oxygen by its carrier molecule, hemoglobin. So while there’s more oxygen in your blood from your deep breathing, less of it is available to our cells.
Fun science trivia: The Bohr Effect was discovered by Christian Bohr, father of quantum physicist Nils Bohr.
So despite a common misconception that you get “drunk on oxygen” during CCB, the opposite is actually true: through vasoconstriction and the Bohr effect, and we enter a state of hypoxia, ie. not enough oxygen.