My understanding, as a non-Muslim who doesn't speak Arabic, is that the standard Arabic is pretty conservative such that MSA, as would be commonly heard on TV or read in the news or other more formal settings, is not very far off the Arabic used in the Quran. So I think most native Arabic speakers would understand the Quran well, but may not be able to fluently speak or write it.
And that many Arabic learners learn MSA along with a dialect, so many second language speakers would probably be able to productively read the Quran as well.
To most Arab speakers, both Classical Arabic and MSA are simply referred to as Fusha.
There are also many non-speakers of Arabic who understand many verses and can pick up on the gist of many verses in the Qur'an.
Alternatively, it may be referring to people who read a language written in Arabic script, but not necessarily the Arabic language. Languages that use some variety of Arabic script include Persian (Farsi and Dari), Urdu, Pashto, Western Punjabi, Uyghur and some other languages. Some of those languages use vowel diacritics, particularly Uyghur.