Suzanna Herculano-Houzel has done some interesting work on comparing the sizes of cells in the central nervous system between various animal species. Turns out that most species CNS cell size scales up with the size of the animal. Primates do this but to a much lesser extent (skipping the discussion on 'power laws' here). It's not so much that humans are special as primates are special. There's some debate over what it was that allowed humans to afford larger brains (ours are 2% by weight consume 20% of available energy - other primates have smaller brains that use less). I lean towards the discovery of fire and cooking food (so externalizing some digestion) hypothesis - but that's all it is - a hypothesis.
That said, the amazing thing to me is that all of us eukaryotes - from protists to elephants to humans - all have about the same machinery in each cell. Plants are also the same, but they've added chloroplasts. Biochemist Nick Lane lays out how this probably happened in his "Why is Life the Way it is?". In there (and his book on mitochondria) is the notion that we eucaryotes have *two* sets of genes in each cell: nuclear and mitochondrial. The mitochondria have mostly given up their genome, so they need proteins both from their own DNA plus proteins from the nucleus to make up critical structures - notably the electron transport chain, where both mitochondrial and nuclear derived proteins fit together so well that electrons can jump from one protein complex to the next - and if they don't... we die.
I'm bringing this up as sepection for perfectly aligned mitochondrial and nuclear genes can explain the difference in lifespan difference between pigeons (about 30 years) and rats (about 3) in spite of the pigeons having a higher metabolic rate. Think about the differing selection pressures in these two animals. Birds need to have good enough respiration to be able to fly, so anything wrong with the mitochondrial/nuclear 'match' means they don't survive to reproduce, so there's pressure for those embryos with a poor match to spontaneously abort. Rats are subject to predation, so it's important as many pups as possible survive gestation and sexual maturity to come quickly. I'm not doing it full justice but check Lane's book for the full argument. IMO very much worth a read.
OK, so *why* do humans live 2.5-3x the time chimpanzees (our nearest primate relatives) do? It's *possible* that one of our ancestor hominids was under selection pressure that somehow perfected the mitochondrial/nuclear 'match'. I'd file that under 'born to run'. Personally, I lean 'grandmother hypothesis', but who knows...?