An equivalent way of saying this, though it needs considerably deeper understanding of QM, is that the uncertainty principle follows inexorably from the properties of the Fourier Transform.
The linkage is that a quantum state is fundamentally a function of complex numbers, whereas in classical mechanics everything is purely real.
Some of you may know why that the question of the existence of uncertainty principle relating energy to time, and why it has the same value as position-momentuum relation is very far from obvious, largely because time is not a quantum operator. A fascinating journey down a rabbit hole finally gives a good answer to that question, but the reasoning is very far from obvious.