As I understood it, their point was that evolution does not add new layers, evolution of the brain always happens by modifying the existing structure.
Which is why a "stratified" view of the brain with evolutionary older layers near the center and newer layers near the surface is incorrect.
The implications of this belief then led to incorrect assumptions about both humans and (non-human) animals: That the human brain is an "animal brain plus something else" and therefore automatically superior - and inversely that animal brains are "human brains minus something" and therefore automatically inferior. The article argues against both positions.