Einstein supposedly once said that you don’t really understand a thing until you can explain it to your grandmother. While I think we can all agree that Einstein was reasonably bright this advice, in its unexpanded form, is fairly stupid.
It encourages people to digest shallow metaphors, maybe memorize a factoid or two from Wikipedia, and then confidently expound upon a subject about which they know literally nothing. I’m sure Einstein wasn’t trying to encourage that sort of behavior, but that’s what’s happened.
What this advice really means is that you should have run the fingers of your mind over the 3-dimensional shape of a concepts so much that you have an intimate acquaintance with its lines and edges. You aren’t just trafficking in facile analogies but can generate a whole host of images, anecdotes, and explanations at will, tailoring them on the spot to better connect with the knowledge already contained in your interlocutor’s head. If they have spotty knowledge of the subject you can skip over those places and drop down in any part of the map that’s still a blank.
Making quantum physics comprehensible to grandmother will not be the same as making it comprehensible to a graduate student in psychology. The grad student might be smarter than grandma, or might not, but that isn’t the only issue. Grandma has a radically different way of understanding the world, a whole host of concepts, intuitions, and biases which can help or hurt comprehension, depending on the context.
She might even surprise you and turn out to remember a good amount of that discrete mathematics class she took 600 years ago.
When you can take the shape of quantum physics in your hands, move it around to expose different faces, change the angle of your explanatory light so that it casts different kinds of shadows onto different kinds of surfaces, illustrate concepts with hand-rolled improvised expositions — with the end result being that your grandmother comes away with a reasonably intuitive grasp of this science, then you understand it.