I find the experiments to be relatively easy to explain to my young cousins (who are planning to take scientific reserach as a career option). I find the theoretical works, especially the ones which require highly involved computational tools (read DFT) to be a bit difficult to explain to these 16 somethings. They have a reasonably good understanding of the structure of the atom but nothing more than that.
I strongly recommend you to visit the site of the North Carolina High School Computational Chemistry Server.
They have a project for teachers and high school students to learn molecular simulations using DFT, Molecular Mechanics and Semi-Empirical methods.
Explaining DFT even to a master's student is pretty hard... and to a high school student? Lol. In that case you're really restricted to a very ambiguous description. Unlike a master's student, they don't know (and you can't show them) what the exchange term is. Moreover, they don't (really) know what a wave function or state is.
Because of this, you're reduced to a very fuzzy and ambiguous explanation, so it would help a lot knowing what context you are working in. Mentioning DFT might even be counterproductive.
The single-particle Schrödinger isn't that hard, and can be understood in terms of the wave equation, e.g. by oscillations of a string. The problem is that if you want to also involve DFT, you need to explain why you need it. This is due to the many-particle aspect: exchange and correlation. The first arises already with a single electron configuration (in addition to the classical Coulomb electrostatics you also get a term corresponding to interchange of the particles), while the latter arises from the fact that in reality many electron configurations contribute to the wave function simultaneously, and the number of possible configurations increases rapidly with system size.
DFT constructs an effective independent particle approximation for this interacting system, and the exchange-correlation functional does the magic of mapping the systems to each other. Although we don't know what the exact functional is, hundreds of approximations have been developed for it, but the problem is that it's almost impossible to tell beforehand whether the approximate functionals are reliable for the system you want to study.
In quantum mechanics state is represented by wave function while physical observables (such as energy, position, momentum etc) as operator. In quantum chemistry we are interested in obtaining energy of molecules, which is obtained as expectation value of the Hamiltonian. DFT is based on the realization that one does not need full information of the wave function as the Hamiltonian involves just one- and two- body operation in the process of evaluating the expectation value of the Hamiltonian, most of the degree of the freedom will integrate out. Thus one-particle and two-particle density will be suffice to obtain energy of the molecule. Two particle density can be written as a product of one-particle densities and a correlation function whose form is not know exactly but can be constructed using physical insights.