Why are the Hamilton cast so famous?

Because Hamilton has permeated American popular culture in a way that no other musical has in the modern age of technology.

I was thrilled to see Daveed Diggs turn up in the Netflix show The Get Down and I think a few other cast members are in it as well.

I think it was similar (though not to quite THAT extent) when Rent first opened on Broadway. The leads were also celebrated and some got great gigs in musical theatre afterwards (Adam Pascal, Idina Menzel...) others went into television, although only Taye Diggs seems to have managed to establish himself there.

We'll have to see yet if the Hamilton leads can turn their fame into a long-term thing or will be forgotten in a few years.

Definitely a pop culture thing that most other shows fail at. With other shows the background perhaps isn't that obvious that the leads came from there, i.e. Lea Michele became a household name through Glee, but she got there after starring in Spring Awakening. Alongside Jonathan Groff, whose career also took off since.

Talent, social media, and savvy PR reps.

But really none of them have fame outside musical theatre. Apart from the guy who wrote it (and I wouldn't recognise him if I found him in my bed) I don't know a single soul who has been in it

So what? Yeah, those who are or were in Hamilton (currently) have niche fame, but having niche fame means... guess what? You haven't heard of them. Some will become famous outside of theatre, some won't. As was already stated, Lin hosted SNL and is composing for Disney's Moana. Even several of my IRL friends who aren't into theatre know Lin as "that Hamilton guy."

And you wouldn't know who they were if you saw them? At some point, every performer you enjoy (or dislike) was new to you.

If you're still trying to say that they're only famous in theatre crowds, you're probably forgetting Lin.

The OP is using a bit of hyperbole, so maybe a better phrasing would've been "Why is the cast of this particular show doing so well with national media exposure that can't be compared to those who've starred in most other Broadway musicals in living memory," rather than using the very interpretive word "famous."

However this isn't Broadwayworld

I would say even Lin ain't famous here. But yeah in the states they prob are and let's not forget it ain't opened here yet so maybe the Hamilton effect will make em stars here too?

He moved to London?