How does a strong electric current harm our body? A strong electric current will posses a great charge. But how does that charge injure us?
High-voltage current arcs (sparks) can make air a conducting plasma, which allows the arc to continue to heat the air, which allows more current to flow, etc. This can produce an explosion called an arc blast which may involve spraying molten metal. The threshold for this is surprisingly low — it's a slight hazard when throwing a 440 V circuit breaker.
Alternating currents can interfere with neuron-to-neuron signaling, causing nasty things like cardiac fibrillation, where the heart muscles lose synchronization and the heart wiggles instead of pumping. By an unfortunate coincidence the frequencies we use for power distribution (50–60 Hz) are especially efficient at stopping hearts, even at fairly low currents.
This Wikipedia entry has some information about this issue.
Basically, the main risk is cardiac arrest, caused by the electric current interfering with the normal operation of the heart muscle.
Other possible damages are burns due to the electric energy vaporizing the water inside the cells, and nerve damage caused by excessive current through the nerves.