"Rapidly" and "quickly" are adverbs that modify the verb "fala", I believe they would be written as "rapidamente".
Yes, this sentence uses an adjective as an adverb, which is grammatically incorrect. But people often do this and phrases like this have become common.
Since "people often do this and phrases like this have become common", the vast majority of linguists (who are linguistic descriptivists), would say that it is therefore grammatically correct.
But it is very literal translation of "Ele fala rápido" which is also not proper Portuguese, because it uses an adjective in the role of an adverb.
"he talks fast" is perfectly proper English. That's how English speakers speak English. It might not be "proper" academic English, but that represents only a small subset of all the different types of English there are, and just like you will look foolish if you speak non-academic English in an academic setting, you will look equally foolish and awkward if you speak "proper" academic English in a non-academic setting. In that setting, academic English is not proper at all and people will look at you weird (not weirdly, weird) if you speak like that.