I really want to use PureScript, but I wonder why I should use PureScript if I already have TypeScript?
Besides, as far as I know, the PureScript part is only for the business logic, not for the rendering, etc like if you want to use React or Angular or pure DOM manipulation. The backend story goes the same (i.e, PureScript only for business logic).
With the extra tooling required (i.e, spago, bower) it seems pretty pointless to use PureScript these days, what am I missing? Someone please convince me to use PureScript.
PureScript isn't just for business logic at all - it has its own UI libraries, React bindings, DOM API bindings, and so on. It is more often used for front-ends than anything else that I've seen.
You only need to use one of spago or bower (spago being the recommended choice now), and that's because npm style dependencies are not suitable for PS.
As for why you'd choose it over TypeScript... well, you'd do so if you want to use a functional language that doesn't allow arbitrary effects and has a sound type system.
I remember when I first discovered it, it drove home the point that choice of data structure can really matter - it made a bunch of problems I'd been working on much easier to solve, as it gave a different way of looking at things.