About half way through my OU quantum mechanics course I realised I had no idea what was going on. I could perfectly well follow the maths: the problem was that I didn't understand how the entities I was learning about connected to 'reality'. The subject might as well have been magick or theology.
I know I was not the only mystified one.
The trouble is that you need to be well-acquainted with the basic framework of QM before you can get its relationship to physical reality straight (it never ceases to be incomprehensible).
So, safe in the knowledge that it makes no sense, here is my current understanding of the basic mechanisms of quantum mechanics.
The quantum state (aka the wavefunction) is what we study in QM; we are not to suppose that this objectively exists out there in spacetime: mathematically, it lives somewhere else, in Hilbert space.
When we propose to make an observation of a quantum system (for example, the energy level of an electron in an atom) we choose the quantum operator corresponding to the property of interest (energy, momentum, position, etc). Call the operator A and the quantum state psi.
If the state of the quantum system can be written in the form A(psi) then we can also write it as a sum (superposition) of A's eigenfunctions, each multiplied by its specific expansion coefficient and the corresponding eigenvalue.
The eigenvalues are the values of possible observations, and the modulus of their expansion coefficients gives the probability of observing that value.
What does it mean to apply an operator to a quantum state?
Suppose the operator is the Hamiltonian for the infinite square well potential, H. Initially the electron had some prior quantum state - maybe it was a free particle. Applying Schrodinger's equation for the infinite square well case to the quantum state psi is equivalent to putting the electron into the box. So applying the operator defines a constraint upon psi which effectively sets up the experimental situation.
However, once we have registered the observation of a specific eigenvalue, the state vector changes again to the corresponding eigenfunction.
So here, in all its incomprehensibility and weirdness, is the reason why quantum mechanics is so hard to grasp, and why no-one really understands what's going on .. even though it all works so brilliantly in practice.