Here I want to set down my understanding of Martin Heidegger's understanding of how Angst is essential to philosophy.
Take a traditional philosophical problem: how do I know other minds / the external world? Sceptically phrased: how do I even know they exist?
Heidegger's idea is that this question will only seem intelligible to someone mired in forgetfulness and taking-for-granted-ness.
His thought is: We've become inured to the contingency and vulnerability and dependency of our lives. Of our sanity. Of our bodily continuity. Of our capacity to think.
And when we are forgetful in these ways we tend to imagine that we are not dependent creatures.
And then we think that we can, all out of our ownmost noddles, raise (what are actually insoluble - but thankfully also misguided) questions about the obtaining of an external world and other minds.
The tacit move is: having forgotten how dependent what's in our noddles is on a world - this world of language, communication, embodiment, friends, teachers, tools, culture, history, society - how dependent the intelligibility and purport of our discourse is - discourse including our philosophical questions themselves - we imagine that we can take for granted the intelligibility of unanchored, free-floating philosophical questions about an 'external world' or 'other minds'.
We take our philosophical questions too seriously because we've narcissistically overlooked our dependency.
If we hadn't lost sight of our essential en-world-ed-ness, it wouldn't have occurred to us to (try to) raise questions about 'how' or 'that' at such an essential juncture. The questions would rightly look both dumb and pretentious.
Put it this way: if our contingency and dependency and embeddedness is grasped as a condition of possibility of our having intelligible thought, then the idea of asking such philosophical questions will come to seem like an absurd scandal.
Yet it's really not easy for us to hold on to the fact of our dependency. We like to think of ourselves as self-contained and invulnerable. We like to imagine we can guarantee our sanity or knowing from within, from an invulnerable procedure of 'pure reason' or from the self-ratification of Cartesian self-presence. We don't like the fact of our dependency - because it makes us angsty.
This, then, is the value of angst. It keeps us on our existential toes. Angst discloses the fundamental condition of personhood as Dasein. The condition - to put it gnomically - of being-here which is being-there.
We philosophers like to think we are bravely asking about intelligibility, about how to secure it, how to achieve it, how to ground it. And all along what we're doing is presupposing that the mind which questions intelligibility can quite happily and intelligibly ask its own philosophical questions. We arrogate to ourselves an extraordinary and utterly unrealistic degree of intellectual sensibleness. We are Humpty Dumpties imagining that our words mean just whatever we want them to mean. What hubris!
Acknowledging our contingent enworldedness is both terrifying and wonderful. We wake to wonder as we acknowledge our smallness, vulnerability, mortality. We wake to gratitude when we acknowledge our dependency. We come to an apt humility, but also to an apt platform - one resting in the midst of our historical-bodily-social lives rather than on a fantasy of a ground beyond them - for exercising in a real, rather than phantastic, and responsible way the power we actually do have.
Angst, I believe, is the flip-side of narcissism. Neither, it seems to me, is necessary (here I depart from many an existentialist), but both are fairly inevitable. In living our lives we tend to unwittingly accrue comforting yet delusional narcissistic certainties. But their smugness keeps out not only the darkness but also the light. Hence the move to humility and wonder, to love and meaning, and to the inevitable angst that comes from relinquishing the narcissistic phantasy.