What's behind the moment of inertia and other "body-global" properties of bodies? I'm an electrical engineer currently doing some (computational) mechanics stuff.
In introductory literature about mechanics, you can read plenty about the moment of inertia and how you use it in dynamics and how to compute it. But so far, I did not find an explanation about the very fundamentals. 
For example, Wikipedia just explains how to calculate it or how to use it to compute the movement of a body under a torque. BUT a real world body does not know about $I$ and $M$, it will just see some (local) forces acting on it. An infinitesimal small volume in a larger body also does not know anything about such body-global properties.
Is there a good explanation about such very basic mechanical principles? (To some extent, I suspect that Monsieur Lagrange had some ideas in that direction.)
 A: Think of a single particle with mass $m$ at $\vec r$ from a coordinate origin. Now suppose this body is in circular motion with angular velocity $\omega$ around this origin. The momentum of the particle is then 
$$
\vec p = m{\vec v} = m\frac{d\vec r}{dt}.
$$
The velocity is then $\vec v = \vec\omega\times\vec r$. Now let us consider the angular momentum of this body $\vec L = \vec r\times\vec p$
$$
\vec L = m\vec r\times\vec r\times\vec\omega = m(\vec r\cdot\vec r)\vec\omega.
$$
The last step is a vector identity. So we have $\vec L = mr^2\vec\omega$. The term $mr^2$ is the moment of inertia of this mass moving in a circular path about the coordinate origin.
Now suppose I have two masses in a circular orbit around this point at different radii, but the same angular velocity. The angular momentum is then just a sum of these. I then in general have a sum of many of these
$$
\vec L_{tot} = \sum_{n=1}^N m(r_n)r_n^2\vec\omega
$$
For the moment of inertia I now convert the sum into a Riemann sum and integrate over a continuous solid body.
A: It is not clear to me exactly what you are asking for : a derivation such as given by Lawrence B Crowell, or a recommended reading list either for the mechanics of macroscopic bodies or for the microscopic properties of materials.
Your question "how does it know?" seems rather a naive one for an electrical engineer to be asking, because you must have asked the same question in your own field innumerable times.  
How does an electrical circuit "know" how to split the current between two parallel resistors? It does not "know" about Ohm's Law or Kirchhoff's Rules.  It does not even know about resistance, charge, electric fields and Maxwell's Equations.  It does not need to. These laws, rules, equations and concepts are our shorthand methods of summarising and describing the regularities or patterns of things which happen in the world.  If you are not satisfied with one level of description, such as current being a flow of electric charge (whatever that is), you can delve deeper, eg by analyzing the statistical motions of electrons in metals, or solving Maxwell's Equations.  But that is really just switching one set of concepts and rules (resistance, current, Ohm's Law) for another (charge distributions, electric fields, Maxwell's Laws).
It is the same with mechanical properties of matter.  The rotation of a solid object can described using macroscopic concepts of the rigid body, moments of inertia and angular momentum, or (as you seem to suggest, and as Lagrange and Laplace suggested) it can be analysed in terms of the microscopic, local interactions between constituent atoms.  But again this is really only switching from one level of description and assumptions to another.  You might be more satisfied by the microscopic description, which is potentially more accurate, but not necessarily any better as a tool for calculating and predicting.  Nor even for describing, since you always have to accept some concepts as "self evident".
