How does mg2+ block the effect of extracellular calcium on neurotransmitter release?

Magnesium is a strong anode than calcium. So, Mg will be oxidized into Mg2+. So instead of having Ca being the reducing agent, Mg will takes its place. That's how the "Blocking" occur.
Without Ca2+, no neurotransmitter was released because the exocytosis of the synaptic vesicles is dependent upon Ca2+.
So, when magnesium is added to the extracellular fluid it blocks the calcium channels and inhibits the release of neurotransmitter.