I just saw Pagliacci last month on one of the Metropolitan Opera's HD presentations, so this is particularly timely. The thing of it is, I don't believe Caesar was considered a highbrow, elitist comedian. Literate and intelligent to be sure, but at the same time there's a lot of slapstick involved in his bits. As well, you probably remember his famous send-up of This Is Your Life that he did on his previous series, Your Show of Shows with Imogene Coca, which indicates his proclivity to satirize conventions with which the audience would be familiar.
And that brings me to my point, which is that the costume, the pathos of the story, all the trappings we see in this skit - they're as iconic to opera as the image of a large woman with pigtails, a horned helmet, and a breastplate and shield. What's more, they're images that people know even if they don't know much of anything else about opera. The television audience - the "middlebrow" audience of which Terry Teachout frequently writes - would have been expected to recognize these images, to know the gist of what Caesar is lampooning. Far from being incomprehensible, the skit was written and performed in order to entertain, to make people laugh - and that entertainment quotient depends on a general familiarity with the premise. The television audience of the mid-'50s would have had that familiarity. Would mainstream audiences today? I doubt it. That's unfortunate; not only do we lose a good amount of comedy because of that, the topical comedy we do get comes from an incredibly fragmented society, targeted not to a general audience (I don't think such a thing exists anymore) but to a very small niche. And today's niche for opera humor - well, you've heard the one about the number of angels on the head of a pin, right?
The writeup of this skit at by the person who posted it at YouTube is very good; take a minute to read it if you can, as it shows just how it matches up with the actual opera. And if you're interested in seeing the actual Pagliacci (a wonderful piece, by the way; I don't think the Met production did it justice), you can see it in its entirety here. Don't be afraid - it's a short opera.