Arnold Kling presents it as an under-popularized idea, so here is my attempt to popularize it:
Let's say you had two people on a desert island, John and Tom, and John wants jazz music on the radio and Tom wants rap. Furthermore any decision procedure must be consistent, in the sense of applying the same algorithm to other decisions. In this set-up (with a further assumption), there is only dictatorship, namely the rule that either "Tom gets his way" or "John gets his way."
Arrow's axioms, by invoking "the independence of irrelevant alternatives" (read: pairwise comparisons) require that all information in the problem be ordinal. A decision procedure such as: "Turn the radio (and allocate other resources) to which station makes the winner happier" is ruled out per se. Which person is happier cannot be expressed in the language of pairwise comparisons, which allow you to claim only that "John prefers jazz to rap" and so on. "Let Tom and John trade" also won't do, whether or not transactions costs are high. That's not a social welfare function as Arrow defined it.
I sense the above paragraph isn't a very charming popularization (it wouldn't make the Freakonomics movie), but let's continue.
That's a two-person example. With 17 persons, Arrow's clever method of partitioning shows that conflicted decisions ultimately must break down to one-on-one comparisons of different individual preferences, returning us to the island/radio example.
For a truly popular audience, you could go on about the difference between inter-profile and intra-profile social welfare functions.
I believe that a popularized version of Arrow's theorem, which explains where the main result actually comes from (strong IIA), is not very impressive to most people. That may be one reason why the theorem doesn't get popularized so much.
Here is one very good transparent version of a proof, which requires virtually zero mathematics. If you work through this paper, you will understand the theorem and where it comes from. Here is a related journal article.