What is the meaning/significance of the values measured from an observable? I'm reading Nielsen and Chuang, and I'm trying to understand what we actually mean when we talk about observables and their relationship to measurements. I'm more of a mathematician than a physicist, and I think there are intuition disconnects when it comes to the intuition/meaning behind these concepts.
When I think about a state $|\psi \rangle = a|0 \rangle + b|1 \rangle$ that gets measured, I imagine that we have a lab and some equipment that measures some property of a quantum object, which causes the state to collapse to $|0 \rangle$ with probability $a^2$. In particular, we can explicitly measure/observe/infer that the object is in state $|0 \rangle$ or $|1 \rangle$  when we use our measurement equipment.
Now, when we talk about a measurement observable we have some Hermitian matrix $M=\sum m P_m$, where $P_m$ are matrices associated with the spectral decomposition of $M$ and $m$ are the eigenvalues. When we "measure" with respect to this, what does this exactly mean? I understand that each eigenvalue is associated with the resulting state $|\psi' \rangle \propto P_m |\psi \rangle$, but what does this mean physically? When we measure our quantum object's state with our lab equipment, we're not measuring the eigenvalues of a matrix, are we? What does it mean to get back an eigenvalue as the result of a measurement, in lab-and-equipment language?
 A: So yes... if you measure the energy of a state, your apparatus records a number which is eventually mapped to one of the possible eigenvalues of the Hamiltonian of the system (these eigenvalues are the possible energies of the system).  Maybe this apparatus actually measures the energy of some radiation emitted in your system, or your apparatus measures a current passing through a coil... whatever.  Some kind of permanent recording (it cannot be erased) is made and through calibration or other means you infer from this a value of energy. Presumably, the outcomes $\lambda_0$ and $\lambda_1$ would generate sufficiently different currents in your coil, or emit radiation of different energies to be recognized as different by your apparatus.
You then redo the experiment and maybe you get a different value, i.e. your apparatus this time records a different current or whatever.  And you repeat lots of time (in theory).  So how are you to model the different possible outcomes?
In your example you would do so by constructing the operator
$$
M=\lambda_0\vert 0\rangle\langle 0\vert + \lambda_1\vert 1\rangle\langle 1\vert \tag{1}
$$
so that
\begin{align}
\langle \psi \vert M\vert\psi\rangle= \lambda_0\vert\langle \psi\vert 0\rangle\vert^2+\lambda_1\vert\langle \psi\vert 1\rangle\vert^2=
\lambda_0 \vert a\vert^2+\lambda_1\vert b\vert^2\, .
\end{align}
The left hand side is the average of $M$ for a system prepared in $\vert\psi\rangle$, and the right hand side is the average of the possible outcomes, weighed by the probabilities $\vert a\vert^2$ and $\vert b\vert^2$ of each outcome.
Thus in this sense the hermitian operator given in (1) reproduces your experimental data: $\langle M\rangle$ agrees with the average value of energy for your data.
Note that a more sophisticated argument involves so-called pointer states and the entanglement of the state describing the apparatus with the state of the system.  The canonical reference for this is the text by Asher Peres:

Peres, A., 2006. Quantum theory: concepts and methods (Vol. 57). Springer Science & Business Media.

