There are things in mathematics that I find difficult to explain even to someone who finished freshman calculus and linear algebra courses. How should one explain singular homology to someone without strong background in topology, algebra and pure math itself? Same thing with theory of sheaves, for instance. These things are not terribly difficult themselves (but they're no easy, I admit), but the amount of time required to explain what they're really about is breathtaking.
I don't know anything about those so I can't say. Not everything has a facile real-world analogy, and there may be long chains of dependent concepts to get through first.
Here's the thing, though: if you make an honest effort to explain something like that to a lay audience, you may fail. Or you may not. Or you'll give them a workable, but incomplete and strictly wrong idea. Either way you yourself will end up with a deeper understanding.
A huge part of "real" mathematics is finding isomorphisms. Teaching is more or less finding isomorphisms between new concepts and concepts that the student already has.
