I come at the issue of Reincarnation as a gentle skeptic. As far as I am concerned, people having detailed memories of things and places they have never seen before is something I have experienced myself, and it is both odd and pretty cool. On the other hand, I would be loathed to suggest that a single answer such as Reincarnation covered all possible reasons for such a thing. Consider genetic memory with some sort of olfactory triggered response, or deja vu in reverse, or dream precognition posing as a lived experience. It's a license to riot. Any half-baked idea is as good as any other without reproducible results.
I note however that nobody has made a case for the Buddhist version of Reincarnation.
The Buddhist version of reincarnation differs substantially from the Hindu version in that it contains the doctrine of Annata. This is the notion that there is no permanent and underlying soul behind the experience. The notion is that with each life you take on "Karmic accretions of experience" that change who you are. Consider the following quote from "John Dies at the End":
"Let’s say you have an ax. Just a cheap one, from Home Depot. On one bitter winter day, you use said ax to behead a man. Don’t worry, the man was already dead. Or maybe you should worry, because you’re the one who shot him.
He had been a big, twitchy guy with veiny skin stretched over swollen biceps, a tattoo of a swastika on his tongue. Teeth filed into razor-sharp fangs, you know the type. And you’re chopping off his head because, even with eight bullet holes in him, you’re pretty sure he’s about to spring back to his feet and eat the look of terror right off your face. On the fol ow-through of the last swing, though, the handle of the ax snaps in a spray of splinters. You now have a broken ax. So, after a long night of looking for a place to dump the man and his head, you take a trip into town with your ax. You go to the hardware store, explaining away the dark reddish stains on the broken handle as barbecue sauce. You walk out with a brand new handle for your ax. The repaired ax sits undisturbed in your garage until the next spring when, on one rainy morning, you find in your kitchen a creature that appears to be a foot-long slug with a bulging egg sac on its tail. Its jaws bite one of your forks in half with what seems like very little effort. You grab your trusty ax and chop the thing into several pieces. On the last blow, however, the ax strikes a metal leg of the overturned kitchen table and chips out a notch right in the middle of the blade. Of course, a chipped head means yet another trip to the hardware store. They sell you a brand new head for your ax. As soon as you get home with your newly-headed ax, though, you meet the reanimated body of the guy you beheaded last year. He’s also got a new head, stitched on with what looks like plastic weed trimmer line, and it’s wearing that unique expression of “you’re the man who kil ed me last winter” resentment that one so rarely encounters in everyday life. You brandish your ax. The guy takes a long look at the weapon with his squishy, rotting eyes and in a gargly voice he screams, “That’s the same ax that slayed me!” Is he right?"
If you are a Buddhist, the answer is "yes" due to the doctrine of Annata. The idea being that if you have a bundle of sticks, and you take some away, and add others, it is the same pile, but if you remove all the sticks, then the pile ceases to exist. The same goes for your soul. Each lifetime you maintain certain appetites from former lives, but lose memories unless they are encoded into your appetites or environment in a way that allows their recall. Buddhas get the perk of enlightenment of recalling their past lives apparently btw. Any RPGer would regard that as a dump-load of free XP IRL if Buddhist Enlightenment is real...all those skills! I guess that makes me a spiritual materialist to covet that huh?
Where this idea gets interesting... really interesting for parapsychology... is that Buddhist reincarnation answers that nagging problem about ghosts that goes "So ghosts have no physical eyes, no retinas, no visual cortex, and no senses that we can physically identify, so how is it that they apparently can see people and interact with them?" Well, Annata tells us that they will have the Karmic accretion of a physical memory of vision that stays with them after death, and these Karmic sense accretions are why people probably reincarnate as people not insects most of the time. The more consumed you are by your appetites, and the more primitive your outlook, the more likely you are to have an animal rebirth. Just remember that a sense of humor is the main Karmic accretion that stops animal incarnations... that is not a joke btw... its Buddha's official statement on the matter.
Anyhow I think that is pretty accurate as a summary of Annata and Buddhist reincarnation, but please do your own research on the matter. I'm no expert.