The best way I've heard MDMA therapy being described is this - you have a large group of extremely painful neurons/memories, and going there is shocking and horrific to the point that you begin to grow a big moat of fearful neurons/emotions next to anything that connects with them. The point of therapy in general is to 1. Break down that fear moat (which frequently grows to be more painful than the underlying issue), and 2. Start drawing those painful neurons and memories out, but in as positive and understanding an environment as possible. At that point the memories/neurons get re-stored, still painful but with the added context of the environment they were last accessed.
MDMA takes step 2 and adds the most positive neurological/chemical setting possible for you to pull those painful neurons and memories out, and puts them in what can be described as a comforting blanket. It can absolutely quicken the process and help you eventually neutralize the traumatic and painful bundle of memories. Calling it a quick fix is such a misnomer - it is incredibly hard work to heal one's self from the inside, no matter the external helpers.