Why do you die if you cannot breathe?
Short answer
This is a difficult question to answer. As far as I am aware, asphyxiation results in excitotoxicity, which causes unconsciousness, brain damage and eventually, death.  

Background
Asphyxia is a condition of the body that occurs from severely inadequate oxygen supply, or because of excessive carbon dioxide in the body (First Aid and CPR courses). The brain is the organ most sensitive to hypoxia (Medscape). Nerve cells in the brain can survive only up to four minutes without oxygen (First Aid and CPR courses). Consciousness is lost after approximately three minutes (Forensic Pathology). Permanent brain damage begins after approximately 4 minutes without oxygen, and death can occur as soon as 4 to 6 minutes later (Medline). Asphyxial deaths typically involve respiratory arrest with bradycardia / asystole (low heart rate / cardiac arrest) because of the hypoxia-induced dysfunction of the respiratory centers in the brainstem (Forensic Pathology).

Despite the small size of the brain (2% of body weight), it is, however, the largest consumer of total body oxygen (20%) and glucose (25%), which are delivered by 15% of the total cardiac output (Schur &amp; Rigor, 1998).

A lack of oxygen may result in so called hypoxic-ischemic encephalopathy (HIE), i.e., neuronal cell death due to hypoxia. Its pathophysiology is related to the lack of energy because cellular respiration diminishes. This initially causes neurons to stop firing, and eventually in an arrest of cellular functions and cell death. Even sublethal HIE can set in motion a series of toxic reactions that kills injured neurons and even neurons that have not been damaged during the initial insult. Thus, following global brain ischemia, neurons do not die suddenly or all at once. In some of them, damage develops hours or days after the insult. Most neurons undergo necrosis. In some neurons, HIE triggers apoptosis (Neuropathology).

Specifically, energy depletion is believed to result in a failure of the Na+,K+ ATPase, leading to depolarization of the neuronal membrane (Fig. 1). Synaptic function and conductivity cease at this point. Depolarization causes neurons to release glutamate (Glu) into the synaptic cleft. Glu is the most common excitatory neurotransmitter. In small amounts, it is indispensable for neuronal function. In excessive amounts, however, it is neurotoxic. Some Glu receptors, such as the NMDA and AMPA receptors, are non-selective cation-permeable ion channels. Initially, over-activation of these channels causes a passive influx of  Cl- (and Na+) into cells causing osmotic (cytotoxic) edema and rapid death (Neuropathology). 

Additional structural damage develops hours or even days later as a result of Ca2+ influx into neurons through NMDA and AMPA receptors (Fig. 1). This delayed cell death is caused by an over-activation of NMDA and AMPA receptors by the excessive release of glutamate, which causes massive influx of Ca2+ into neurons. Ca2+ activates catabolic enzymes (proteases, phospholipases, endonucleases), and also NO synthase. NO is a free radical and other free radicals are generated due to the impairment of oxidative phosphorylation. Free radicals and activated catabolic enzymes destroy structural proteins, membrane lipids, nucleic acids, and other cellular contents, causing neuronal necrosis. DNA damage from endonucleases and mitochondrial injury from free radicals trigger apoptosis (Neuropathology). Collectively, these effects are referred to as excitotoxicity (Choi, 1992). When enough brain cells die, the person perishes with them.


Fig. 1. Schematic showing the excititoxic effects of excess glutamate in the brain. Source: Neuropathology.  

References
- Choi, J Neurobiol (1992); 23(9): 1261-76
- Schur &amp; Rigor, Dev Neurosci (1998); 20: 348-357