I would like to say something about my understanding of this topic to see whether I am on the right track.
First and foremost, as I understand it, the important point is that there is no self, ego or personality which goes from life to life. Instead it is more like a process, where the actions, thoughts and intentions generated here in this life, give rise to a future life which embodies the qualities resulting from these.
As for the cultural origins of this idea, I imagine that in ancient times, people had a much stronger sense of continuity with the past and their ancestors. This is because of the oral culture, where the stories and wisdom were memorized in handed down from generation to generation. Within this context, it must have been much easier to see newborns as like an 'incarnation' of a previous life. In the ancient world, there was a very wide-spread view that everything happens in cycles of creation and desctruction - people, seasons, tides, cultures, and everything else. Within this view of the world, re-birth seemed entirely plausible.
Here in our modern world, everything is changing so quickly, and we are so remote from our ancestral past, that we seem to live in this kind of bubble of the present, surrounded only by space, and preceded only by the 'blind' forces of evolution.