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: 
All that, with no conductor in sight!

I'd put it down to possibly not enough rehearsal time, or an inexperienced orchestra. There's always this sort of intonation/tuning problem with either. Playing in tune with other strings takes years to get right - playing one string instrument by itself in tune is enough of a challenge, and they did really well with a challenging piece.

It's true that choirs and string orchestras sometimes do move slightly 'off key', particularly the more amateur choirs, when there's no reference point like a piano to follow, but the better ones will all tend to move all together, rather like tempo changes with no conductor to keep time. But your 'off key' I guess is meant for 'out of tune with each other' - something subtly different.
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: