What they're getting at there is something that is fundamental for chemists and physicists, namely the Ehrenfest theorem. This theorem basically says that quantum mechanics is broader, more general version of classical mechanics (so like Lewis acids/bases being more general than Bronsted acids/bases) and thus quantum phenomena reduce to classical observables when you're dealing with a classical system. At large principal quantum numbers, the system is classical and thus you would expect the system's properties to also reduce to what you would expect from classical mechanics.
In terms of electron orbits, classical mechanics predicts a spectrum of absorption/emission - there no "disallowed" transitions. However, in quantum mechanics, electron orbits are separated by some finite energy distance, like the rungs of a ladder instead of a ramp. So according to Ehrenfest's theorem, when electron orbits are viewed classically (i.e. at high principal quantum number), the distance separating each successive orbit becomes smaller and smaller until electrons look like they can slide up and down any distance they want on a ramp. As you go to smaller and smaller quantum numbers, this ramp resolves into a ladder, as per quantum mechanics.
This is a physical explanation of this phenomenon and you can even look up Ehrenfest's derivation if you're interested but one can also make a simple electrostatic argument as well. Say n = 1 corresponds to a radius of 1 and n = 2 corresponds to a radius of 2 and so on. Doubling the radius of the electron then results in a quartering of the attractive force between the electron and the nucleus. It takes a lot of energy to reduce that attractive force. But if you're only moving the radius from 10 to 11, the attractive force doesn't change by very much. Thus, you don't need as much energy.
