The trick here is that they start with the intuition of counting, in which each number is itself a single thing, but are actually doing a calculation based on each number being a string of single digits.
It's much less shocking if you say: Picking a random 10 digit number is the same as randomly picking 10 single digits. The more digits you pick the more likely it is that you'll get a '3' somewhere in there. So as your string of numbers increases in length the less likely it is that you're string won't contain any given number. If I said to someone "The more dice you throw the more likely it is you'll get at least one 3", I don't think I would get anyone who was surprised by that.
The unfortunate thing is that math and number theory in particular are full of genuinely fascinating observations that don't rely at all on a tricks of phrasing to be revealed, and rather than making less math literate people feel 'dumb' (as these videos tend to do), spark in interest in exploring math further.