Your basic understanding is correct. Of course, there is no such language as 'broke' - the meme caption is a joke. "You don't have airpods? Sorry. I don't speak broke" Together with similar memes, this meme relates to the standard assumption that airpod owners are rich show-offs ( 'braggarts' is the word you'll probably find in your dictionary). The joke here is the perceived cultural gap between airpod owners (wealthy) and non-airpod owners (poor). It's suggesting that airpod owners see themselves as belonging to a world so different from non-owners that they don't even speak the same language.

It isn't clear, and it isn't a normal thing anyone would say. And no, 'broke' is not a language. You are right that that might be what the person might be saying, but as I said, it's not clear. Do you have a reference? More context?