It's definitely bullshit, but is perhaps derived from observations people have made.

Most power sources have limits on the amount of amps they can drive. If you try to pull too many amps, the voltage you see will roll off.

Take for example a AA battery. It's rated for 1.5V. Let's say you short the battery, connect 0Ohms across the terminals. Well I = V/R, so I = 1.5/0 ... which means I is infinite, right? Infinite amps!

But of course that's not true. In fact, the battery has internal resistance. Most of the time, it can be ignored, but at high amperage it becomes significant. In the case of a AA battery, you can only get at most about 10A out of it. The voltage just keeps rolling off; you'll never get more than 10A out.

So, getting back to the point "it's not volts that'll kill you, but the amps?". There's some truth to that. All that really matters is how many amps you put across the heart. You may grab a battery rated at some insane voltage, do the math based on the internal resistance of the human body, and calculate that it'll put enough amps through your heart. But maybe the battery can't actually deliver that many amps, because of the battery's internal resistance. So if you were to touch that battery with both hands on either side ... you'd live. The voltage would slump and prevent enough amps from crossing your heart.

There's also the question of AC versus DC. Your skin has a higher resistance to DC currents than it does to AC currents. So it takes more DC to kill you than AC.

In practice, though, don't follow the adage. If the volts are high, GTFO. Best not gamble on your understanding of electricity, ya know? Leave it to the professionals, like ElectroBOOM: https://www.youtube.com/watch?v=dcrY59nGxBg