Having thought for some decades about how to convey the substance of "the underlying mathematics" of second-generation mathematics and beyond... sadly, I think it is not really possible (without lying, etc).
That is, even the genuine basics of contemporary mathematics in 2018 are several steps abstracted/evolved from the otherwise-intuitive physical mathematics that (I might claim) most human beings understand instinctively. E.g., just by using visual cortex and being able to throw and catch a ball, and/or not crash cars much at all, considering the high volume.
I might claim that a large part of the disconnect between contemporary mathematics and the general (intellectually/scientifically inclined) human population's context is due to the nearly-compulsive mathematical style of backforming concepts from examples and applications. Apart from perhaps conspiracy theories, this is not what most people do. E.g., to compare a doughnut and a coffee cup? Srsly? Cute, at best, and what's the point? No, that trope is worse than unconvincing... it is convincing of the frivolity and goofiness of mathematicians. And, of course, topology was never motivated by the homotopy equivalence of coffee cup and doughnut. This truly is fake news.
By this point, I have come to think that the best way to justify/explain the point/efficacy of various bits of "pure mathematics" for non-mathematicians is to explain the outcomes. Even though I am sympathetic to Lockhart's Lament, it doesn't "sell" to lots of people, and, in fact, I think accidentally parodizes (for a good purpose, I know) professional mathematicians' motivations. Just as G. H. Hardy's "Apology" did, I think.
So, seriously, don't turn math into some weird steampunk fetishism... (No offense intended to those people and their hobby!) All the "gears and stuff" in math are not just decorative. They are functional. And not just Rube Goldberg's version of "functional".
So, yes, sadly, many of the technical details do not lend themselves... Not everything can be sold in an elevator pitch to laypersons...
(So, to be clear, I no longer try to explain the "genuine" underlying mathematics, but only the manifestations "on the surface". People understand those phenomena much better, because they fit into the standard phenomenology of human culture.)