The philosophy of Krishna Consciousness is fundamentally rooted on certain core tenets one of which is the concept of "transmigration of the soul from one body to another" or in common jargon called "reincarnation".
Reincarnation is a sensitive subject matter for almost everyone as the generally accepted theory is that we have only one life and hence live to the fullest gratification. Majority scientists identify the self to be the body and mind. Judeo-Christian faiths advocate only one life and then follows eternal heaven or hell. Most if not all deal reincarnation with high degree of skepticism as it is very difficult to prove empirically.
Lord Krishna elaborately speaks about this in the Bhagavad Gita some 5000 years ago and clearly supports that the soul is eternal and the body is temporary. After we die, we take another body depending on the current set of actions we perform in this body. Therefore it is paramount that living beings live a life of dharma to get a good future body, better yet live a God centric devotional life to break the cycle of birth and death and return to the Kingdom of God.
Below is a video that presents reincarnation as an empirically verifiable scientific fact than religious dogma. If we approach this concept temporarily keeping aside our preconceived notions, perhaps this video will reveal a truth. Extensive research has been conducted by the renowned and Late Dr Ian Stevenson on this topic for the past 30 years and more information can be found here.
I am happy to say from a philosophy standpoint this topic has already been dealt in a holistic manner by Krishna ages before. Anybody interested in this phenomenon, in my opinion, should read the Bhagavad Gita As it is written by Srila Prabhupada.
Please watch the video