The audio sample is pretty understandable. Obviously the speaker took great care to enunciate clearly for the recording, so he sounds like he speaks very understandable Swiss German with a lot of French vocabulary thrown in. Like a French exchange student or the like.
I think given the title, "the Swiss language that few know", they should have picked Romansh. The fourth official language of Switzerland that got completely overlooked by the article. A descendant language of the spoken Latin language of the Roman Empire. Haven't met many people around the world who have heard of it.
I agree the sample is not difficult to understand as a nativ Swiss-German speaker. Only thing that comes to mind is that it's a Bernese (?) Swiss-German dialect which is not always easy to understand for 'proper' German speakers.
But in my opinion almost every single dialect in Switzerland has it's own specific words and almost every canton has it's own dialect so I wouldn't call them a language.
[0] http://www.bbc.com/travel/story/20180627-switzerlands-myster...
My guess is, that it's be most famous example city for the eastern parts, although for British people Klosters would be a known alternative as well, their cable car booth is name "Prince of Wales" and in the Prättigau they rarely speak Romansh
I have a friend speaking Le Bolze. When he is around Swiss Germans, he still speaks it, altough in a much less pronounced fashion. He uses much more German words and only occasionally the odd French word.
The dialect of Valais is a lot harder to understand than this. But I guess in normal conversation it's a different case again.
Another interesting language is (rather, was) "Matten-Englisch", something like Pig-Latin, "because people wanted to communicate in a way the police would not understand": https://en.wikipedia.org/wiki/Mattenenglisch