It is not likely that any part of the brain evolved specifically to support reading. Identifying which parts (or structures) of the brain support specific mental processes is a challenge, in part because any mental process typically depends on several different brain structures, and because each brain structure typically supports different functions. But in many cases it is possible to get insights into the evolution of some brain function, such as visual perception or memory consolidation, by studying the brains and behaviors of other species, such as monkeys and rats, and piecing together what we know about the evolutionary history of these species. Typically, brain structures emerge as adaptations to a novel evolutionary pressure over hundreds of thousands or even millions of years.
Neuroscientists have identified a region of the human brain that seems to be particularly apt to identifying written words. It is called, not surprisingly, the "visual word form area", and it is located in the underside of the brain's left hemisphere (the left ventral temporal lobe). However, this region also seems to participate in the processing of other kinds of visual stimuli, and it is difficult to say to what extent it is really specialized for identifying words, and not any familiar complex visual pattern. What we do know, is that writing only appeared in human history within the last ten thousand years or so, and until relatively recently, was restricted to just a select few members of the human species. Therefore, reading has not been an evolutionarily relevant force (i.e., a selective pressure) for long enough to have caused changes in the genetic blueprint of the brain. Another compelling argument against there being any parts of the brain that evolved specifically for reading is the fact that the ancestors of modern Native-Americans and Australian Aborigines, among many other ethnicities, never -- ever -- developed any form of writing, but a Native-American or Aborigine child can learn to read as easily as a child whose ancestry has been reading and writing for millennia.