However, how did they engineer things so that they had one? Probably a big part of the answer was that they created an external womb in which the human brain could be better incubated.
This external womb took the form of baby slings. These slings created a protective environment for the human cranium after birth - allowing more premature birth to take place - before the skull bones had fully formed. The result was a marsupial-like phase of human evolution - with the newborn living in a synthetic pouch.
The next-most significant factor letting memes inflate the human cranuim was probably reduced nutritional constraints - a hypothesis covered in some detail in the book The Driving Force: Food, Evolution and the Future by Michael Crawford and David Marsh.