Electron shell, how do they know it?
Electrons are not actually separated in shells. That is a simplified model to explain the atom's structure when people first start approaching Chemistry or Physics. 

In reality, electrons' movements around the nucleus are governed by complex equations (some of them not even resolvable so approximations are needed) to form what is called an electron cloud. This cloud is confined in proximity of the nucleus by the electromagnetic force. Every electron is defined by its wave equation, which tells us the probability that the electron appears to be at a particular location when its position is measured. These equations allows us to define regions in space where the electron spend the most part of its time around the nucleus, and they can be visualised as atomic orbitals, as shown below. Orbitals can have one or more ring or node structures, and they differ from each other in size, shape and orientation.



Each orbital (which is defined by a different equation) has different associated parameters, called quantum numbers, and a particular energy. These properties allow us to divide them into levels and sublevels, to form a structure that resembles multiple shells one inside each other. But as you can see from what I explained the structure is in fact far more complex.

This knowledge has been gained over time both from the mathematical and physical analysis of different equations (like Schröndiger equation) and by experimental data collected by the means of different techniques. Starting from Thomson and Rutherford the model of the atomic structure has seen many changes (Bohr, Moseley, Van der Broek, de Broglie, etc.) and it has come closer to a realistic representation of the actual atom. Take a look at their experiments to understand how it has been studied. However, they often involve particle physics and are not really straightforward to understand.