If I had to explain large cardinals in 10 minutes my first source of inspiration would be Kanamori's excellent writings about their (whiggish) history. People like history. I wouldn't spend any time trying to justify or prove any result. Maybe 2 minutes to explain the cumulative hierarchy so that everybody understands the typical picture of V. If people guess there is a long coherent tradition of research starting from Cantor, hence of interest in the field, I think that's enough. They really don't need to understand anything, just have a general feeling of a sort of flow of ideas.