Heisenberg's Uncertainty Principle is, in essence, a consequence of two basic facts: de Broglie's relation between a particle's momentum and its associated wavelength, and a mathematical fact known as the bandwidth theorem, which states an equivalent uncertainty relation between a wave's position and its wavelength.
The fundamental physical leap here is of course the first one, the statement that particles have a 'wavelength' of some meaningful sort. This leap takes you directly outside of classical physics and there is nothing within the classical sphere that is really enough to motivate it. De Broglie's relation remains a cornerstone of quantum theory, though it is usually stated in a more sophisticated form as the canonical commutation relation $[\hat x, \hat p]=i\hbar$.
The bandwidth theorem, on the other hand, is a very intuitive statement about waves, and it essentially says that if you want to measure a wave's wavelength $\lambda$ to some uncertainty $\Delta \lambda$, you will need to measure a number of periods $n$ that is inversely proportional to $\Delta \lambda$. Thus, on one end, to have zero uncertainty in the wavelength you need to measure an infinite number of periods, and on the other end, a wave pulse that's so short it only fits one peak and no troughs can hardly be said to have a defined wavelength.
The precise statement is cleanest in terms of the wavevector $k=2\pi/\lambda$, and the theorem states that the size $\Delta x$ of the region occupied by some waveform, and the width $\Delta k$ of the region of wavevectors that appreciably contribute to the waveform's plane-wave decomposition, are related via
$$\Delta x\,\Delta k\geq 2\pi.$$
Since de Broglie's relation reads $p=\hbar k$, the extension to the Heisenberg principle is obvious.