From a brain imaging study and analysis of the auditory regions and brain pathways in humans, apes and monkeys, published in Nature Neuroscience:
They discovered a segment of this language pathway in the human brain that interconnects the auditory cortex with frontal lobe regions, important for processing speech and language. Although speech and language are unique to humans, the link via the auditory pathway in other primates suggests an evolutionary basis in auditory cognition and vocal communication.
Professor Petkov added: “We predicted but could not know for sure whether the human language pathway may have had an evolutionary basis in the auditory system of nonhuman primates. I admit we were astounded to see a similar pathway hiding in plain sight within the auditory system of nonhuman primates.”
Press Office, “Origins of language pathway in the brain at least 25 million years old” at Newcastle University
Paper. (paywall)
But surely the most remarkable part of the story is that of all these life forms, only humans speak.
See also: The real reason why only human beings speak