The word angst, meaning neurotic fear, anxiety, guilt or remorse comes from German, but has been used in English since the 1940s. It comes from the same root as the word anger, and was originally popularised in English through translations of the work of psychoanalyst Sigmund Freud.
The word has more recently adapted however, and now teenagers are much more likely to talk about feeling angsty. When they complain about not being understood by anyone, or that they are alone in the world, they are feeling angsty. When they insist that you could never know how terrible it feels to them, how the whole world wants to destroy them, they do it because they are feeling angsty. An angsty person feels completely unloved, unwanted, disrespected, ignored.
Some cruel people might suggest that those who feel angsty are just looking for attention. But these people don’t understand us. And they never will!