Well, for one -- it's good.
Don't get me wrong, I haven't spend the thousands of dollars to buy a ticket to go see it, nor do I live in New York where I can just take in a popular Broadway show.  But simply by listening to the soundtrack, I can tell that Lin-Manuel Miranda is some sort of modern lyrical and musical genius.
The weird part is that I didn't really want to get "swept up" in the phenomenon.  Every podcast that I was listening to had people raving about this new musical that utilizes rap and hip hop to tell "a" story of our founding fathers.  I mean, call me old -- but I don't really like the modern take on those musical styles.  I can absolutely respect the modern artists' talent, but it just isn't my thing.  So when I heard that this show utilized this type of influence, it seems that I set my preconceptions to further not be interested.
Then I listed to the crazy thing.
Stinking amazing, man.  It's poetry.  It really is.  It's catchy.  It touches a cord and does more than just make me bob my head while listening.  It may be that secretly I like hip hop and rap more than I think.  It certainly is not that I'm a powerful history student.  
It evokes emotion where it needs to -- affects me.
And that's really what we all want, right?  We want our writing to produce something.  A feeling.  A reaction.  But not just a reaction for reaction's sake.  I think it has to do with story.  When the lyrics tell me the story with a fantastic and powerful style and fit together with such passion and energy -- it's a win.
Not to mention that there is just enough "musical" in it to bring me smiles when I think of the big shows that I have been a fan of in the past -- (all the King George songs in particular.)
This Miranda is a pure artist -- and the show is almost TOO good.  It's like he hung out too long at the talent counter and picked up something that belonged to someone else.  So as a writer, it leaves me with that wonderful conundrum:  the exciting desire to produce something as good as Hamilton -- coupled with the devastating frustration that I'll never produce something as good as Hamilton.
Back to story, though.  The way Miranda plays with stanzas and words is awesome and inspiring.  It wakes up that little sleeping poet in some of us writers and lets him wave his little hands in the air for an hour while we listen and enjoy.  
If you haven't listened to it, you owe it to yourself to just test it out.  All it takes is the first song to give you a taste -- then on to some other great examples like "My Shot," "Right Hand Man," and "Washington on Your Side."
Love it.  Hope he tells us another story someday soon.