This pattern of brain growth in humans is a reflection of the delayed maturation cycle in humans. Humans mature much slower than do other primates. Remember our baby teeth are not replaced with permanent teeth until we are much older. This delayed maturation is more than offset by the increased intelligence associated with a larger brain capacity. The human intelligence is able to deal with a variety of environmental problems because the greater intelligence.
Humans are neither strong or fast but have come to dominate the Earth.
It may not be hard to understand why it took over 1.5 million years for Humans to evolve after the appearance of the Australopithecus. Remember the Australopithecus spent a great deal of time in the trees. A mother Australopithecus could not climb and care for a helpless infant while climbing trees. The rapid maturation of the Australipithecus infant was important therefore to the survival of the mother and child. This would not be possible for Homo mothers since their infant is not able to fend for itself until much later in life.