In my opinion, the best explanation of the theorem is not to get into the details right away but to say simply: it's why you and your coworkers always have trouble deciding where to go to lunch.
As I read the discussion the key idea I learned is that societal preferences are not transitive.
If voters cast ballots as follows:
- 1 vote for A > B > C
- 1 vote for B > C > A
- 1 vote for C > A > B
then the pairwise majority preference of the group is that A wins over B, B wins over C, and C wins over A: these yield rock-paper-scissors preferences for any pairwise comparison. In this circumstance, any aggregation rule that satisfies the very basic majoritarian requirement that a candidate who receives a majority of votes must win the election, will fail the IIA criterion, if social preference is required to be transitive (or acyclic). To see this, suppose that such a rule satisfies IIA. Since majority preferences are respected, the society prefers A to B (two votes for A>B and one for B>A), B to C, and C to A. Thus a cycle is generated, which contradicts the assumption that social preference is transitive.
Yeah, the terms may not make sense without reading the whole article, but they key point is that even if individuals have transitive preferences, the aggregation of those preferences will not necessarily be transitive.