Dodo - I can understand your perplexity, but this is a central tenet of Buddhism. The Buddha's teaching of the Middle way says there is no soul - as this would be eternalism - the survival of something from a person's life -(I think in our romantic ideas of reincarnation we think of a persisting personality). The Buddha also says that there is no annihilation - which would be the sceptical/ scientific/ materialistic view.
Instead he posits a radical view that it is the impetus from one life which causes another through Karma. The classical analgy is of a candle flame lighting a second candle flame. The first is blown out leaving the second flame. Clearly the second flame is not the first - no persisting soul - but it is caused by the first, and so there isn't annihilation.
It is a question which is central to Buddhism and the understanding of Karma and reincarnation. It is a much more subtle idea than our usual take on reincarnation, and requires study and investigation.
There is another thread on reincarnation where we are looking at this idea too.
The question of evidence is difficult, because of the definition of what constitutes evidence. Clearly my subjective experience is not evidence for you, and I can't see how evidence of reincarnation can be objectively provided. There is a Professor Stephenson who conducted research into reincarnation cases in children, but these also suffer the same problem. So why should i believe it? I am a rational person brought up very much in a scientific tradition with atheist parents.
One reason is that following a Buddhist path has meant that I have been able - in a minor way - to test out some of the teachings. My thinking is - if this teaching is correct, then perhaps the rest of the teachings are too.
There is also my personal experience that leads me to believe in reincarnation.
Another more general reason is Buddhism's approach to teachings which specifically states that teachings should be tested out. This is to try to stop the charismatic leader syndrome, as well as being an honest and confident approach.
There is also the reincarnation of HH The Dalai Lama. In his case, it is not about evidence, but about the quality of the man.
I would disagree with your asertion that religion puts people into groups, though I understand where the idea comes from. People put people into groups generally, and the xenophobia and wars are usually conducted for the national/ tribal gain. Anyone can claim God is on their side, but whether that is the motive is questionable. Wars etc are usually fought over land and power with religion as a pretext.
I decided before I was Buddhist that there was no God, and follow the Buddha's teaching on this, but i do have a lot of respect for the good work that religious people do on the quiet, for no self gain.

