Per the lit I could access from their website (I don't seem to have access to much of the actual lit, this is from their propaganda):
Apparently Mg has antagonistic effects on the NMDA receptor, as well as presynaptically inhibiting GLU release in the first place. I suspect this is the initial reason they looked at Mg specifically, as a good part of stroke research at the moment is the search for a safe and effective NMDA/GLU receptor antagonist to prevent excitotoxicity. I guess they also found that it caused some local cerebral vasodialtion and some other metabolic effects involving inhibition of Ca++ release via blocking the ion channel.
Translations:
NMDA = a synthetic analogue of Glutamate (GLU), binds only to specific NMDA GLU receptors in the brain, to which GLU will bind, but GLU binds to all GLU receptors, not just the NMDA GLU receptors.
Glutamate = excitatory neurotransmitter (as opposed to inhibitory neurotransmitter) which depolarizes a neuron which possesses a GLU receptor (aka moves it closer to firing).
excitotoxicity: over stimulation of neurons by an excitatory neurotransmitter (E.g. GLU acting on NMDA receptors). This is one of the primary pathological processes in stroke, the accumulation of GLU is a result of ischemia and results in overstimulation of a neuron and then neuronal death due to accumulation of Ca++ in the cell.
Thus: the idea is to stop the overactivation of a cell which posesses a GLU receptor in an ischemic episode. This is accomplished by antagonizing (blocking) the GLU receptor (specificially the NMDA class of GLU receptors in this case) and thus blocking the overstimulation of cells, and therefore hopefully saving the neuron.