I agree with a previous poster that from the nature of the question and the level of the
student, the most satisfying answer should involve the ideas of resonance. First because
that gets you out of the mindset of thinking of the electron as an isolated particle which
is important.
The classical harmonic oscillator is a good place to start for resonance, but to get at the
ideas of quantum levels one needs to think about the quantum harmonic oscillator as well.
And the lesson of the quantum harmonic oscillator is that to get something corresponding
to classical swinging or sloshing one needs to have a mixture of states. Recall that the
states corresponding to individual levels are called "stationary states" because they are
time independent. So resonance, or the excitation of motion, involves mixtures of states
necessarily. Hence there is no mystery about "how does the electron know another state
exists".
Another tack that might resonate particulary well (sorry) for a chem student is to think
about what kind of motion a uniform electric field can excite. It wants to separate charges
to create an oscillating electric dipole. So applied to an atom we need to find mixtures of
states that create oscillating electric dipoles. Using pictures we can see that a mixture
of a 1s and 2p state does just that. So not only do we see why mixtures of states are
important, but also a little about selection rules.
I think semiclassical approaches are good because of the intuition they develop in
students.