Observables correspond to particular things you can do in the lab (or observe in nature). So let's first talk about something you can do in the lab.
You can take a particle with spin and subject it to an inhomogeneous magnetic field. A particle with spin has a magnetic moment proportional to the spin and we know to Hamiltonian for a particle with a magnetic moment in an externalmagnetic field. So we get a term in the Hamiltonian proportional to $B_x\hat\sigma_x+B_y\hat\sigma_y+B_z\hat\sigma_z.$ At each point, this is an Hermitian operator. By setting up the magnetic field to point in the $\hat z$ direction and to be inhomogeneous in that direction we can send a beam in that is eigen to $\hat\sigma_z$ and it will be deflected up or down (depending on the eigenvalue and on how we made the field vary in the z direction) and this deflection literally happens because of the evolution determined by the Schrödinger equation when you have a term in the Hamiltonian proportional to $B_x\hat\sigma_x+B_y\hat\sigma_y+B_z\hat\sigma_z.$
And so now when you have a general state and again evolve it according to the Schrödinger equation when you have a term in the Hamiltonian proportional to $B_x\hat\sigma_x+B_y\hat\sigma_y+B_z\hat\sigma_z$ then the incoming spatial state splits its beam into two beams, one going up and one going down and the size of the two beams is such that the total current in each beam has a ratio equal to the ratio of the projection of the spin state onto the eigenstates of $\hat\sigma_z.$ Again, that fact is determined by the Schrödinger equation evolution for he actual state of the subject (the thing with spin) and the Stern-Gerlach device (the thing with the inhomogeneous magnetic field). And the spin state evolves so that the branch that is spatially deflected up becomes spin up and the branch that is deflected down becomes spin down. And again this cones out from the evolution determined by the Schrödinger equation when you have a term in the Hamiltonian proportional to $B_x\hat\sigma_x+B_y\hat\sigma_y+B_z\hat\sigma_z$ then the incoming spatial state splits its
So we know what causes measurements. Devices interacting with subjects according to the laws of physics. The Hamiltonian always has a term proportional to $B_x\hat\sigma_x+B_y\hat\sigma_y+B_z\hat\sigma_z$ when there is a particle with spin and an external magnetic field, there is no choice about whether it is there.
And we know the effects of measurements, they split states into a sun of states. Each each term in the sum has an eigenstate of the operator and that eigenstate is entangled with some other state. In this example the spin state become entangled with the position state of the particle. Deflected up entangled with spin up, and deflected down entangled with spin down.
So the only mystery is why we call it measurement. And that is because of that polarization, now if you put it on a similar device again it won't be split it will just be deflected. The real measurement effect happens later when these different states have a chance to affect the states of many other things so the two branches no longer can interact just because it is too hard to get them to ever overlap again.
So now to the question of operator. It isn't random operators, it is operators that have real eigenvalues and that have orthogonal eigenvectors. That is what is important. Why?
The real eigenvalues allow the continuous splitting of a state into the multiple eigenstates, which is what allows the Schrödinger equation to be able to split the state to entangle the projections with something else in a continuous manner (which is what the Schrödinger equation requires and we have to evolve according to the Schrödinger equation for the actual experimental setup there is no choice about that and no other options). To actually do it well, you have to find something else to couple with the eigenvectors to get the entanglement. And for that you can't actually just do anything, you are restricted to real terms in real Hamiltonians and nature only provides so many to choose from.
So why do we need the eigenstates to be orthogonal. That is key to getting that do it twice get the same answer. Otherwise it is just an interaction, not a measurement.
So to call something a measurement it basically has to be a real linear combination of orthogonal projections onto mutually orthogonal states. And so they do need to be operators, very particular operators. And you can only actually observer them if you can find something else to couple differently to those different states.
It's still not clear to me what the direct meaning of $\hat A |\psi\rangle$ is.
The point is that the different eigenvalues you get from $\hat A |\psi\rangle$ tell the other thing that is becoming entangled with your object to continuous change at different rates depending on the eigenvalue.
For instance if you have a general spin state then for a fixed magnetic field the spin zero doesn't couple at all so isn't deflected at all and the lowest non zero spins deflect but deflect less than larger spins so deflect in opposite directions but at a power rate of motion in the deflected direction. So spin zero goes straight, spin $\pm 1/2$ get sent a little bit up and down and spin $\pm 1$ get sent more up and down. And I mean the z component t being 0, 1/2, or 1 which I why I included the minus sign to make that clear. So the eigenvalue tell you how separated the other thing (the thing used the measure the state) becomes. This is literally why you project onto an eigenspace. When two things have the same eigenvalue the thing you couple with doesn't move differently for the things with the same eigenvalue so they don't get separated at all so the whole projection onto the eigenspaces gets entangled with the exact same state of the thing used to measure it.
It is a wavefunction
The eigenfunctions are the ones that don't change in repeated measurements son get deflected in different ways as a parameterized by the real umber eigenvalue.
how does its corresponding quantum state relate (in physical terms) to the original state $\psi$ and the measurement $\hat A$?
The original state can be decomposed as eigenstates and they evolve into entangled states as they separate. So when separated you have the different eigenstates entangled with different states of the thing used to measure it. Spin is easiest since it entangles with its own position so we don't really have to bring in anything else other than the inhomogeneous magnetic field.