This happened to me two days ago! The "What is it that you teach then?" conversation.
I first said everyone's different and has different brains, and that a bunch of us who did PhDs together decided being a PhD mathematician was a mild form of mental disease, with a lot of common symptoms (that I won't need to list to anyone who's spent a lot of time observing people who teach postgraduate mathematics), and told her to ask my wife about my maths friends at our wedding, remembered by the families for ever.
I explained we all hit a wall in maths at some stage, (the wall being where your innate talent becomes insufficient to get on without a significant increase in effort, reached at very different ages by different people).
With a good teacher, you learn to climb over the wall and start finding wall climbing enjoyable. If the teacher at the time doesn't get you over your personal wall, then because it's so hierarchical (I didn't use that word) and jargon-laden, it's easy to get left behind and conclude maths is impossible and leave demoralised. I went on to explain how I've managed to get my students this year much better at (metaphorical) wall climbing, and they're the best bunch in terms of mock exam results I've had for a long long time.
(Aside: a lot of this is to do with eliminating judgement from feedback and replacing it with instruction, in conjunction with opportunities to improve. The opportunities to improve have always been there, but the culture of not-there-yet, but here's what to change is the cultural difference that has resulted in the behavioural difference.)
She went on to tell me about her personal wall at work (tracheotomy re-insertions) and decided she wanted to do something about it - planning to start observing her colleague doing it and get accredited to do it. Apparently she's felt for more than a decade she ought to be able to do this, but decided to actually do something about it when hearing that it's possible for students who find mathematics hard to get past that! I don't know whether she'll follow through on that, but I'm delighted that some of what I said must have meant something to her.