The non-transitivity of Rock-Scissors-Paper is easy to understand, partly because it's so simple, but mostly because you're likely never played outside the usual rules, even if adding Lizard-Spock.

Non-transitive dice screw with the 'nature' of dice that most of us expect. To get to the mathematical intuition, one may have to get past a deeply-ingrained feeling that something about these dice just isn't right. That's a big part of the fun.