Why do Ion Thrusters require such high voltages but have relatively low currents?
Think for a moment : particle energies are measured in electron volts. 

So to increase the energy output, you either need to increase the number of electron charges (i.e. the number of ions), or accelerate them across a higher voltage.

And Tsiolkovski's rocket equation makes it clear you want to impart the highest velocity (thus most energy) to the smallest possible mass. This is what gives you the highest specific impulse from a given reaction mass, assuming you can generate the electrical energy somehow.

Put this together and the ion current will be as low as possible for a given energy, and that, using electrostatic acceleration, means using the highest practical voltage.

(Even if the acceleration is magnetic, the final particle energy will be measured in eV, and the current emitted in the microamp range, though the acceleration coils may be low voltage, high current devices).

Unlike a chemical rocket where the energy source (fuel) and reaction mass (combustion products) are closely related, the energy per particle is divorced from the nature of the particle itself, and therefore potentially unlimited. This is why ion drives can deliver much higher specific impulse.

It's not rocket science...

EDIT : Nayuki's comment below adds another part of the picture : this process is actually less energy efficient; as you reduce mass you have to increase energy per unit of momentum imparted to the vessel. 

However the tyranny of the rocket equation means (to a close approximation) that doesn't matter; energy is cheap, (solar or nuclear) reaction mass is expensive once you're out of the atmosphere, which is why you need huge tanks of it.

This explains some of the excitement about reaction-mass-less drives such as the E-M Drive or Cannae Drive despite their (so far) monumentally low thrust.

The rest of the excitement seems to be based on their inability to follow the known rules of physics, despite the best efforts of the peer reviewers to cancel out experimental error...

Notes: 


I have some suspicion that the reaction "mass" will turn out to be composed of photons; but that seems to be de-bunked as part of one paper (linked from here)
Skeptics may note that in Scots, "Cannae Drive" means "Can't Drive". We shall see...


EDIT 2 : Re: Tony's question, apparently not all ion thrusters are pulsed, and it's easy to imagine how a continuous thrust electrostatic one would operate. 

I've seen one small "pulsed plasma thruster" in (lab) operation, operating well below 1 pulse per second. One can infer from data here - thrust 40 micronewton-seconds, total impulse 44 Ns, a lifetime of around a million seconds at that power, and a pulse rate about 1 pps. (It can be throttled back so lower rates are possible).