It's popular because it uses all types of musical styles including rap, R&B, and brit pop to tell Hamilton's story. The cast is also racially diverse, the only character that has a cast member with the same race as he is King George, a minor character.

IMHO, it has resonated with people because it takes a stuffy topic and makes it more relatable to younger generations allowing them to see a bit of themselves in the founding of this country. It also goes against a prevailing thought that it's ok to "white-wash"(recast minority roles with white actors) but not the inverse. Also, the internet, this was the viral video that started it all: https://www.youtube.com/watch?v=WNFf7nMIGnE

Unable to score tickets to see the show, fans have coalesced around the really awesome soundtrack.