Reincarnation used to be the belief of many many cultures. The Pope had most of the references removed from the Bible as a political move, but certainly not all. Any acurate history of the Catholic Church will tell you this information.
At that time, it didn't pay to believe in reincarnation because it could end that life very quickly with a big bonfire. For safety, the belief began to die out and over the years people began to accept the new version of the Bible and eventually didn't even remember what had been there.
Past lives can show up in a several ways that you don't realize. Child prodigies, a passion about another culture, dreams, phobias, flashbacks etc. I experienced one of mine when I accidentally duplicated a similar situation to my death in that life. I then remembered how I died and forgiving those who suffocated me unknowingly.
Remembering the whole life isn't necessary, we normally bring those bits and pieces through that are still unresolved, or that will serve us in this life. Those are the ones you work with. The rest isn't there because you don't need it.