Quantum theory basically says that energy can only exist in discrete quanta (basically specific amounts that increase in jumps). This is related to the shells – for an electron to become “excited” (or jump) to a different shell, it has to gain that specific amount of energy.
I’m not sure which experiment you are referring to, but if you mean the one involving hydrogen spectra, it’s basically like this:
He filled a tube with hydrogen gas and shot some electricity at it. The tube then emitted light. However, instead of seeing a nice contiguous rainbow of colours, he saw discrete, disconnected bars of colour. The reason for this is because the electrons in the hydrogen gas were absorbing these specific quanta of energy they needed to become excited to a higher shell. However, electrons don’t particularly like to be excited – they like to be in the “ground state”, or shell number one. Therefore, they immediately hop back down to the ground state and EMIT this energy they absorbed as LIGHT.
Now to the quantum theory: Since the electrons could only absorb specific quanta of energy to become excited, they can only emit the same quanta of energy when they hop back down! This is why only certain colours were emitted from the tube. Each colour corresponded to a different energy lever which corresponded to a different shell of the hydrogen atom.