Why are atoms mostly empty space?

Professor Brian Cox is a physicist in England, very well-known there as a popularizer of science. The reasons for this are many-fold, including his ubiquity across media (including podcasts, Twitter, and of course TV)... but also because he has an infectious enthusiasm for science coupled with a boyish charm. This was all on display recently when he hosted a great segment on the BBC's show A Night With The Stars, where he simply and effectively demonstrates why atoms are mostly empty space: