Does Darwin's Theory of Evolution explain why the human brain is so big?

Darwin's theory specifies descent with modification through natural selection. What that means essentially (and as an over simplification) is that DNA - which we now know to be the engine of change in species - is copied wrong, on occasion, leading to a modified gene. Sometimes those modifications have consequences for the animal. In the case of human brain size, perhaps some early hominid had a gene coding for the additional split of neurons during it's prenatal life (all animals, even ants and beetles, have the same basic brain chemistry. The main difference is that, during development, higher animals' brains are told to keep producing neurons, thus increasing in size to accommodate. Our brains produce the most neurons per overall surface area). So this hominid, and it's offspring, and maybe some relatives all have slightly better brains than their contemporaries allowing them greater powers of abstract thought, communication, and organization...MAYBE. They soon start out competing their neighbors for Natural Resources thereby increasing their numbers though better food, shelter, etc (again, this is all hypothetical and oversimplified). Soon their numbers are so great, they completely overwhelm the competing populations whose numbers crash. They are, unwittingly, moving toward becoming a new species, over thousands of generations, with slightly bigger brains. With the better food (mostly protein) comes the fuel to support the development of such large brains.

The same is true for all modifications from giraffe necks to whale spouts. They developed overtime through a series of accidents that worked in the animal's favor environmentally.