The most basic explanation for the Heisenberg Uncertainty Principle is that the momentum and position of a quantum particle is not very distinct when an attempt is made to measure them together. But what is it that causes the uncertainty? Because if there is no change in the momentum, then it would be the same as measuring the two separately. So what causes this change in the instantaneous perceived momentum. A change in mass? Velocity? Or net composition of the particle?
The Heisengberg uncertainty principle is a mathematical formulation of how nature is in the microcosm: in distances of the order of Angstrom the value of some pairs of observables can only be bounded and not measured to the accuracy we might want.
What does a position measurement mean?
It means setting up an experiment that will measure the location of a particle. The uncertainty principle states that if you want in the same experiment for the same particle to measure the momentum, you cannot measure it more accurately than the uncertainty principle allows. It is a fact of the quantum mechanical nature of reality at the microscopic level ( mostly, there are a few macroscopic quantum effects also).
This sentence in your link:
The particular eigenstate of the observable A may not be an eigenstate of another observable B. If this is so, then it does not have a single associated measurement as the system is not in an eigenstate of the observable.
describes the quantum mechanical situation well. The pairs of observables falling under the HUP cannot both have unique eigenvalues of the system's quantum mechanical solutions .If an observable is not an eigenstate of the system the only constraint on its value comes from the uncertainty principle.
So it is not that p changes, it is that we measure a different px for the particle when we know its x position accurately, and vice verso.