Similar behavior is witnessed in animals often. An animal will learn a novel behavior -- generally some basic tool use, and then its relatives or packmates will quickly adopt the technique. It spreads like wildfire.
They clearly already contained the capacity to perform that behavior, they just needed to be shown that they could do it in the right context. Once they were shown once, they adopted the behavior quickly -- and even pass it on to their young.
I believe early humans developing new skills and learning was similar.
It was not that humans began speaking or writing as soon as it was biologically feasible -- rather some (long) time after feasibility was evolved, the skill was developed and then had to "go viral" and infect other humans.
How many times did it start and stop before it hit critical mass? How many tribes of humans were available for this kind of development to even take place?
Fascinating stuff, there's an enormous amount out there on this subject, and I only have a BS in Biology, so I'm no expert!
EDIT: "Why the sudden jump in progress" You may mean, "Why did human brains rapidly get larger over an evolutionarily short amount of time" -- and the answer is that there was no sudden jump, only slow increase over time (as measured in fossils).