I don't really agree with pretzel's definitive analysis of why small children often speak in terms of remembering a previous life, and neither do I pretend to understand it as a phenomenon, but I do know that it's a recognised developmental stage that seems to occur in all cultures and social groups, and sometimes involves things that almost certainly could not be learned from their own environment (e.g. languages no one at home speaks in communities that don't have electricity, still less TVs). Having had a child who spoke in quite clear terms about his 'before life' (not famous or moneyspinning, sadly) I can also tell you that it's a whole lot harder to dismiss when you hear it with your own ears.
I do believe in reincarnation, but only in the sense that Brian Cox believes in it - i.e. in the sense of the entire universe and everything in it being constructed from the same matter that is endlessly constituted and reconstituted into trees, animals, diamonds, sludge on the sea bed, bananas, people, etc. I suppose hypothetically, if a larger than usual amount of matter found itself by great coincidence sequentially reused in first one person and then, later, in another, it might account for past life memories like this, but...I don't know, I don't know. It's a fascinating phenomenon though, to be sure.