Let's face it, life is tough, and it's not made easier when you're a chew toy.
And when the going gets tough — and let's face it, since True Art Is Angsty, it will — even the tough can start to whine. Of course, depending on age, the whining will be different, but nonetheless they'll whine. In fiction, however, no matter how justified the character's complaints are there's a delicate balance between enough whining and too much, and unfortunately characters that should have a right to angst can lose the sympathy of the audience if the audience no longer wants to listen. Many situations in fiction do not happen in real life, therefore most audiences cannot truly sympathize with the character and can find them annoying. This phenomenon is known as Angst Dissonance.
Note that Angst Dissonance is a personal limit, and different people have different levels of tolerance for exactly how much they're willing to put up with; furthermore, a lot of these depend as much on how the character is being read/written rather than any inherent issues in the text.
Essentially, as noted above, this trope largely exists due to a prevailing belief among some that True Art Is Angsty; that 'art' should only be concerned with exploring angst and pain and thus (in theory) reflect the real world. Of course, most people engage with art and fiction at least in part to escape the problems of the real world in a fictional one where these problems can be resolved more easily and satisfactorily, and don't always like having them thrown back at them (especially if, as noted above, the creators are going way over the top with it). These creators also tend to ignore/forget/disdain the fact that life is also made up of uplifting, encouraging bits as well, which people like to be reminded of. This can cause problems to arise between the creators, who are attempting to make an artistic statement about their characters and the world they inhabit, and the readers, who simply find that artistic statement too depressing (or annoying).
Angst Dissonance can lead to creators going right into Wangst if the Dissonance goes way over the top, as no matter how genuine the complaint people will get turned off if the character's moaning goes too far, especially if all they're doing is complaining without attempting to effectively deal with the problem. If the creators try too hard to generate angst for the characters without accurately judging exactly how much angsty backstory the audience can reasonably tolerate, and don't take steps to pull back when they might be going too far, then Deus Angst Machina can be the result. Angst Dissonance can also be the result if the show's relying on a visit from Diabolus ex Machina, which can damage the show's credibility and the audience's ability to suspend their disbelief. If the character's complaints stem from something that is actually pretty cool, then the character is Cursed with Awesome — and audiences hate it when characters complain about powers and abilities that they'd love to have.
Using a Cozy Voice for Catastrophes can mitigate Angst Dissonance, in that while the character in question is still hurting deeply, at least he's pleasantly chatty about it. Similarly, the Happy Flashback can show that characters weren't always angst-bags.