I'd assume it means their primary language is Italian, and they basically only interact with other Americans who's great grand parents were Italian and primarily speak Italian.

So like a snapshot of Italian culture as it was a few generations back like how Quebec is a snapshot of historic France and Newfoundland is a snapshot of england

My great-grandparents migrated from Italy to Brazil. Is there a better way to phrase that?

It's not a really strong identity, we don't cultivate the ancestry like Americans do, so unfortunately I don't speak the language, neither do most of the ~30M italian descendants in the south. The deeper into the countryside you go though, you start hearing old dialects of italian (and german).