The phrase "it's the current that kills, not the voltage" is vastly oversimplified and ambiguous.
<PERSON>' answer details why current flowing through your body is a bad thing, so I'm going to focus on why I dislike this phrase.
For starters, to get a current to flow through your body in the first place, a sufficiently high voltage is necessary. After all, humans aren't incredibly conductive, so the electrons in your body aren't going to start moving without a reasonable potential. That's why you don't have to put on a bunch of protective gear every time you change your car battery-- even though it can easily output hundreds of amps, its paltry 12 volts can't push a dangerous current through your body (Also, as a word to the wise: low voltage, high current supplies can still cause injury. My electronics teacher was always fond of telling a story about how his friend was working on such a supply while wearing a ring, and at one point his wrench on one terminal touched the ring which was contacting the other terminal, and the resulting high current through the ring heated it up and caused him to lose the finger). So, my correction #1 to the phrase would be "but a voltage is required to induce a current".
But even this isn't the whole story, as different voltage sources behave differently under load. 75
For instance, static electricity isn't usually dangerous because even though the voltage is high enough to push a significant current through your body, it very rapidly decreases as it discharges, and the time span is short enough to avoid any real damage. But if you go out and grab two different phases of a 480 VAC mains power source (note: don't do this), the line to line voltage will pretty much be unchanged by your addition to the circuit, and it will continue to push the same current through you indefinitely. So my addition #2 would be "Current alone does not guarantee lethality; the duration of this current and total energy delivered to your tissue also play a crucial role. Since different power sources behave differently under load, the danger of a given setup must be analyzed on a case by case basis."
As you might be able to tell, writing pithy rules of thumb isn't my strong suit :)
EDIT As freecharly pointed out, current in the body is primarily carried by ions, not electrons, and the low conductance has to do with how many of them there are. But the main point of the paragraph still holds: you can't induce a dangerous current flow through the body without a high enough voltage. 75