In the standard (Copenhagen) reading of Quantum Theory what is occurring is that reality as it truly is cannot be imagined or conceived of by the human mind or modelled scientifically.
All we can do is arrange a situation where the fundamental "stuff" will leave a trace in our macroscopic world. This trace will always be comprehensible as a classical property since it will be evident in some device we use.
Thus the position and momentum referred to in Heisenberg's uncertainty principle are actually positions and momentum of the device, not the particle.
The uncertainty principle then shows you that these properties cannot be combined. The results you get from position measurements and momentum measurements do not sensibly combine. You can see a mark in a detector, then measure momentum and get "moving to the left" then measure position again and it is to the right of the original mark. This is to be expected though since position and momentum are not true properties of the microworld.
What quantum theory lets you do is predict the chance of the next mark/trace in a device given the input of a previous one you saw. That these are random is part and parcel of the fact that you don't have a theory of what is actually going on and never can have such a theory, since the true reality lies outside human conception.
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