Two things make that terribly funny. First off only Eric Idle could tell that story with that smirking straight man approach he has. He has the comedic confidence and wit to know the story is funny with a simple and direct delivery. Classic Python humor: regular stuff is tremendously funny when delivered with the wink and smirk of irreverent logic.
Second Idle is such a comic virtuoso that he knows not to add punch lines to the story. He simply tells it straight and lets the thick irony of the situation be funny. The lack of punchlines is so "German" and Idle uses that to add a layer to the inherent ironic humor of the situation.
As a final note, this is a classic example of why us nerds revere the Python's humor. As with all Python humor, Idle treats the audience like they're smart enough to get the joke. He doesn't beat the audience over the head with the humor or guide them to the laughs. Instead he expects that everyone in the audience is smart enough to get the joke without dumbing it down for them or cluing them in. They never give it to you and if you don't get it then you're the ultimate butt of the joke. The people who don't see the humor in how terribly silly the whole of mankind is thinking and acting are ultimately the ones they are telling jokes about.
When someone tells me they don't like Monty Python I tell them that is because they are the one's the Python's are making fun of. The joke is on them because they are the cool kids who think everything is terribly serious, and if you take anything so seriously you can't find any humor in it, the Pythons are the eighth grade school boys that point and titter. However the reason it is so brilliant to me is unlike the rest of us nerds, the Pythons so cool they could get away with laughing at the establishment and the cool kids would lap up the derision heaped on them by the Pythons and ask for more.
The point of Monty Python's humor is not only to point out that the emperor has no clothes on, but also to point out the absurd conspiracy by the rest of the world of pretending they don't notice the emperor has no clothes.
Son, you've got a panty on your head.