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: 
The other answers have given lots of reasons why classical musicians might go off pitch. But in this case, most of the issue is simply as Todd Wilcox and Tim wrote: the orchester is inexperienced, they didn't mean to play out of tune, still it happened.

But I would say a bit more about that, more as an apology on behalf of these musicians than anything.

First: specifically for the cello introduction but also the rest

– Well, to begin, this is a high school student who has perhaps never played in front of an audience of the size they were facing here. And then she has to begin this piece *all alone*, with literally every eyeball in the room upon her. Easy to see how that can affects one's performance. (Maybe not so much for the kind of personality who plays lead guitar in a metal band, but orchestra musicians tend to fall a bit more on the shy side.) Even nerves aside – string instruments are tricky enough to play accurate under the best of conditions, but when you've just taken your seat, maybe the position isn't quite right, maybe there's something wrong with the chair, etc. etc..

Professional musicians have the experience from hundreds of public performances to guide them through such adversities, and even they get it wrong sometimes, and/or they're hyper-sensitive about the temperature backstage and whatnot because they know the dangers.  
(Well, and there's also musicians who are super-relaxed badasses that just step on stage and kill it every time, but it's a big skill.)

Second, intonation isn't everything.

We're living in a world where so much music is hyper-corrected. Leaving aside the question of whether the predominant 12-edo tuning is even a good ideal – often other aspects of a performance are far more important than ideal pitch. This piece is a *rock song*. For sure an unusual one and in a different arrangement, but it doesn't change the fact that it's much more about powerful expression and rhythmic energy, than about clean intonation. And that's frankly something many orchestras totally suck at, even professional ones. MHS is doing a pretty good job at it I have to say, at least until about 2:30 where the tempo starts dragging a bit (probably bow arms getting tired). Even in the melody, there's a lot of simply anger and desperation. This can involve phrasing, effects like portamento (sliding/bending into- or out of notes), but also simply a certain not-giving-a-damn about intonation.

In summary: these musicians did the best job that could realistically be expected from them in the situation.
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: