Question: Can anyone explain to me what's going on?:

I am not and have never been a professional classical musician, but I don't think you need to be one to see that something is definitely wrong here — starting from the very intro by cello and then throughout the whole piece. And I am more than assured that performers themselves realize that. And that's what puzzles me greatly. They, nevertheless, perform it as if it were exactly the way it is supposed to be. And it doesn't seem to be any kind of joke on their part either.

Is there any more or less reasonable explanation to this? This act reminded me of my first experience of having one kind of French cheese. It smelled like nothing less than a piece of well-rotten remains of a corpse. However, before I was ready to vomit, one good person explained to me that that was a special French delicacy and was, in fact, good for health. Well, while I was still doubtful about how it was good for health, at least I got some explanation.

So, is there any similar plausible explanation for this case?

Answer: 
Another example of deliberately playing out of tune I'd like to bring to the table is perhaps one of the most stunning pieces on the classical guitar: Nikita Koshkin's *Usher Waltz*. This piece was inspired by the novel *The Fall of the House of Usher*, and the composer deliberately wrote out of tune passages to showcase the absurdity that the story has fallen into. Here is an example performance, and the out-of-tune "Bartok pizzacatos" start at 3:50:
Yes it is a thing. If you go into modern practice of old music (baroque and prior) it is quite common to see intentional out-of-tuneness for effect. And even more that that slight shifts from intonation can influence the character of a piece and are thus rather common. But in this case this is simply because these are *not* fully trained classical musicians, this is a high school orchestra.

This is the actual recording of the string quartet this arrangement came from: