For example consider Japan. Nearly every region in Japan has its own dialect. Everyone learns, for lack of a better way to say it, "standard Japanese" which is the kind used by news broadcasters. But, to their friends and family in their hometown they speak their local dialect which is often not understandable by people outside their region.
They know when to speak standard Japanese (for example a job interview) and when it's okay to speak the dialect.
The article is suggesting that AAVE should be considered a dialect and treated the same way. That seems reasonable to me given it's the same in many other countries. It also means respect for the culture of AAVE instead of contempt which seems like a good thing to me. So many people speak it. Why is their culture any less valid than another?