No oxygen means no final electron acceptor, meaning no ATP generation. No ATP generation means pumps like Na/K pump don't work (as one example). This depolarizes the cell because the gradient can't be maintained. Depolarization leads to glutamate firing (it's an excitatory neurotransmitter in the brain). This is something called "glutamate toxicity" or excitotoxicity.
This in turn leads to increased cytosolic calcium, which can activate endonucleases, phospholiases, proteases, etc. and punch holes in the mitochondria. This leads to leaking of cytochrome c, which activates caspaces.