There’s an old saying among power company engineers and electricians that goes “it’s the volts that jolts but the mills that kills”, meaning that high voltage may make you jerk more, but how much current flows through you (the milliamps, what they are calling the “mills” here) is what determines whether or not you die.
So in a sense it is the current that kills and not the voltage, but the two aren’t unrelated. Generally speaking, the higher the voltage the greater the current.
What he said was that you can’t have high amperage though a large resistance without high voltage. Low voltage and high current implies a very low resistance. You are missing the resistance part of that, which is very important in this case.
For simple ohmic devices, the formula is V=IR, where V is the voltage, I is the current, and R is the resistance. If the resistance is fixed (i.e. it’s a simple ohmic situation) then the current is proportional to the voltage. The only way you get more current is to increase the voltage. So the voltage and current are not unrelated. If you have a high resistance and a low voltage, then the resulting current will be small. That’s just the way it works.
A 12 volt car battery can crank out a whole bunch of amps, but you can safely grab the positive terminal with one hand and the negative terminal with the other and you won’t get shocked, because the voltage is too low and your body’s resistance is too high. Once you get above 50 volts or so though, then the voltage is high enough that significant current starts to flow, and you can get shocked (and killed).
One important thing to mention at this point is that the human body isn’t ohmic. We don’t have a simple resistance. A common (if way oversimplified) electrical model for the human body is a resistor in series with both a resistor and capacitor in parallel. The thing is, the values of those components changes according to the applied voltage. If you grab both leads of a multimeter set to measure resistance, you’ll measure somewhere between several hundred thousand and several megs of ohms of resistance. That’s because the battery voltage which is being used to measure your body’s resistance is very low. At high voltages (like a few thousand volts) your body’s effective resistance drops to maybe a few hundred to a thousand ohms or so. A lot more current flows through you at higher voltages, much more than a simple ohmic prediction from a multimeter would lead you to believe.
As for what actually kills you, electricity tends to kill you in one of two ways.
The first way is that it screws up your heartbeat. If your heart gets hit with a shock it can go into fibrillation, which is a funny kind of mode where it’s rhythm is all out of whack and instead of beating it mostly just sits there and shakes somewhat chaotically. Your heart has kind of a funny design in that this fibrillation state is actually stable, meaning that if your heart goes into fibrillation it tends to stay there unless something disturbs it (like someone whacks your heart with a shock from a portable defib machine). Since your heart isn’t effectively pumping blood in its fibrillation state, you pass out within a few seconds, and die shortly thereafter.
It takes surprisingly small amounts of current to throw your heart into fibrillation, and this is current dependent, not necessarily voltage dependent (though again, the voltage often determines the resulting current so they aren’t unrelated). Most safety standards in the U.S. are built around anything under 5 mA being generally safe for the heart. Once you get above 5 mA though, you’re in the danger zone, and that’s not a whole lot of current (5 mA = 0.005 amps). Admittedly, at 5 mA the heart isn’t terribly likely to go into fibrillation, and the chances increase with the current. At around 50 mA the heart is much more likely to go into fibrillation, and that’s still an awfully small amount of current.
The thing is, it’s kind of hit and miss. Your heart is much more sensitive to disruption at certain parts of its cycle than others, so a lot depends on exactly when the shock hits you.
Generally, as you increase the current, the heart becomes more and more likely to go into fibrillation, but then a funny thing happens. Once you reach enough current, instead of going into fibrillation, the heart muscles just tend to clamp. At that point the heart isn’t pumping blood, so if someone doesn’t remove the source of electricity from your body pretty darn quick you are still going to be in a whole world of hurt, but once the current is removed, the heart will usually go back into a normal rhythm. So it’s kinda odd that as the current increases, as this point the fatality rate actually starts to decrease.
But then as you continue to increase the current, you get into the second way that electricity kills you. It literally cooks you to death. And now, as you increase the current, the fatality rate once again climbs. If you were to push a nail into each side of a hot dog and then connect each nail to a wire, then plug that into a standard wall outlet, you’ll cook the hot dog in no time (there are safe ways to demonstrate this, which involves having the whole thing on a wooden base so that you don’t actually touch anything while the electricity is applied - it used to be a favorite of kid science shows way back when). The electric chair cooks you to death in the same way. Lightning also kills you by cooking you to death. Cooking you to death is mostly a function of the electrical power that gets absorbed by your body, which is both the voltage and current (P=VI).
While low level currents throwing your heart into fibrillation might be a bit hit and miss, cooking to death generally isn’t so random. The survival rate of a perfectly functioning electric chair is very low.
