As mankind made its evolutionary transition from ape to human, scientists think that there was a time when a flushed cheek and a downcast eye was a key survival instinct. When one species was still evolving into the other, it meant your genetic line wouldn't end in a dismembered pile of limbs. Blushing would have been especially useful, since it's an involuntary reaction. No matter how hard you tried to maintain eye contact, your blushing face broadcasts just how close you were to soiling your loincloth.
Blushing, after all, signals remorse. Since it can't be faked or hidden, researchers think that our ancestors evolved the function as a display of appeasement towards those that we've wronged. Or who could kick our ass. The walk back to the cave might have been shameful, but it was accomplished with fully intact sexual organs -- the only victory that matters when it comes to the gene pool.
The part of the species that averted their eyes and turned beet red no matter how much they wanted to stand up to the big aggressive guy taking their lunch lived to see another day, and breed with the (presumably underwhelmed, but realistic) females of the species. The big guy eating our lunch might have won that battle, but that same unwavering belief in himself was what sent him out in a blaze of monkey-fisted Neanderthal asswhoop. The meek end up inheriting the Earth by default.