> The mystery is in saying that there is actually "no voltage", and the 
> current is just magically flowing all by itself.

Well, the quick way to explain is to look up the formula for an inductor:

V(t) = L * dI / dt

So, the voltage is proportional to the rate of change of current.  This is 
because a magnetic field is generated by the flowing current, and when the 
current changes the magnetic field generates a reverse-voltage to try and 
cancel out the current change.

Nowhere does it say that a voltage of zero means no current flows.

If you've got 5 V and 5 A flowing through an inductor and you then suddenly 
switch the current to 10 A, it's quite possible that the inductor will 
generate a back-emf of -5V (so giving a sum of zero across the inductor) to 
try and keep the current at 5A.  During the split second that the voltage is 
zero, it is behaving exactly like a piece of wire with no resistance.

Or for a capacitor:

I(t) = C * dV / dt

This says that the current is proportional to the rate of change of voltage. 
Nowhere does it say that no current can flow if the voltage is zero.  But, 
note, the rate of change of voltage can't be zero, otherwise no current 
would flow.

But for a resistor:

V(t) = I(t) * R

In this case, I=0 means V=0 whatever is going on (for non-zero R)

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