You absolutely cannot explain the details of a research question in ten minutes, but you should definitely be able to explain the philosophy -- in the literal sense of "love of wisdom" -- of your problem in that much time. Mathematics lets you take really primal ideas (like continuity, symmetry, smoothness, shape, proof, truth, size, chance and information) and make them precise. This is really wonderful, and it's very much to your advantage to be able to communicate this wonder to others -- this is really what distinguishes mathematics from accounting or chess.

Anyway, in ten minutes, you can't communicate all of this, but you should be able to show off a gem or two. The go-to subject for this kind of demonstration is group theory, but if you want to focus on your own area, you have a big advantage. Large cardinals belong to logic, and so numerous great mathematicians have spent an enormous amount of effort trying to understand and explain them.

For example, you can motivate large cardinals by giving a safe impredicative definition, such as the greatest lower bound of a set, and then contrasting it with some unsafe ones, such as Russell's paradox or the Liar paradox. This tension -- between the enormous practical utility of impredicative definitions and their tendency to open the door to paradox -- can be used to motivate large cardinals via the question of how close to the edge is safe.

Note that the question here is really a basic one: when does a definition actually define something?

A good resource is Poincare's essay "Logic and Mathematics, II", in which he argues for predicativism. Large cardinals are sort of the maximal rejection of his view, but Poincare is such a good writer and thinker that he's worth reading if only to get the most beautiful exposition of the alternative.