The article isn't discussing accents, but rather differences in vocabulary and grammar between two dialects. There is a difference "He isn't going to do anything about it" with an accent and saying "He ain't gon' do nothin' bout it."
Interestingly, double negatives were standard in English up until about 200 years ago, when English speakers wanted their language to be more like Latin-- that is, civilized and refined. So they changed it.
An interesting part of AAVE is that is has verbs conjugated by tense and aspect: https://en.wikipedia.org/wiki/African_American_Vernacular_En... . Aspect is something standard English doesn't have a grammatical case of.
For example, what's known as the "habitual be".
In SAE, one would say something like "Sam is walking that route to the store and does so regularly." In AAVE, an equivalent statement would be "Sam be walking to the store that way."
The habitual quality of the action is what would allow for the use of the habitual be. One would not say it for an activity that is only expected to happen once.
To further illustrate, in SAE you'd say "He's dying." In AAVE, you would not say "He be dying." because it's not habitual.
There are many others but I think this is one of the easiest to explain.