The reason this explanation is so widespread is that it is actually not false. However, the actual uncertainty in the Uncertainty Principle takes place at a much deeper existential level, in a more-important way, so this is a woefully incomplete explanation.

The actual uncertainty comes from the fact that the two quantities are Fourier transforms of each other... and just by that relationship, inherently, if one gets very localized (== very high frequency bump in its space), its Fourier dual gets spread out very far through space. (You can sort-of analogize this if you know about audio ... a sharp spike in temporal space, when transformed into frequency space, becomes a very big spread of values, because all those frequencies are relatively blunt and they have to somehow fit together to make this sharp thing, which requires an enormous number of them. Or if you go the other way, a sharp spike in frequency space means one frequency, which transforms into an infinitely-long sine wave in temporal space. So think about that kind of thing, except instead these are waveforms where the y value is kind-of the probability of getting that particular x-value as a result if you perform a measurement.)