'Teen Angst' (or 'Teenage Angst') is a term-couplet bandied about by many people from all walks of life, from the obviously naive to those who think they aren't. It's also very poorly understood and used too-often in a condescending manner; usually by those not 'teens' anymore.
So, basically, teenagers don't really 'get' it because they're living through it; and non-teenagers don't get it because they're not teenagers any more. Psychologists don't get it, because they're asking the wrong questions, and philosophers don't care, because they don't care about anything that smacks of being less than something 'mature'. Parents don't get it because parents, by and large, are hopeless at parenting, and especially those aspects that require putting themselves into their offspring's emotions. Media people don't get it, because they also don't really care about anything but the notions of dimwitted adults about what it's like to be a pre-adult. Educationalists don't get it, because they are obsessed either with controlling or 'managing' the phenomenon—it being very inconvenient to them, as might be expected.
And yet Teen Angst is not only essentially simple, but indeed an element of 'growing up' that the likes of me would consider essential for creating a rounded, aware and emotionally developed human being.
It is nothing but the juvy version of the same angst faced by every more 'mature' Absurdist or Existentialist—an angst that has its roots, as you may have guessed, in our awareness of our mortality, and which is especially pronounced in those who, for whatever reason, are unable to live in permanent denial-mode about the dreadfulness of our ultimate extinction. Teens—Young Adults!—usually do not yet have the sheer body of life and experience 'context' required to deal with it in the manner 'adults' would. And since they haven't learned yet to displace and prevaricate about it, it all comes out in those symptoms that we know so well, and some of which may be very dire indeed.
And, yes, it is as simple as that. Add to that a brain developing toward some form of neural maturity, plus a hormonal system in a state of acute turmoil, and...well, that's what you get. Nothing mysterious about it at all. And certainly nothing that requires the condescending, paternalistic attitude displayed toward it by adults; many of whom will deny their own former experience of the phenomenon, because they feel it's almost embarrassing that they were ever subjected to such silly emotional upheavals. And it also is nothing to fear by those undergoing it, and I wish the apparatchiks running our educational systems realized that the best way to deal with it is not to deny or try to 'manage' it, but to help those in their care to welcome it with open arms and accept the dread it brings to their lives.
It is through facing dread that we grow. Pretending it doesn't exist isn't only cowardly, but, above all, simply stupid.