Atman is a pure form of Brahman but entering the Prakriti it gets various attributes (Guna) like Sattva, Rajas and Tamas. Hence, this Atman falls into a life Cycle of Birth and Death. As a human changes his clothes and wear new clothes but it doesn't changes the identity of that human, same way we human have many births. Atman enters a body, stays in it until he dies and than moves on to a new body. This is the process of Reincarnation and it happens when Atman enters Prakriti (Nature) or Sansar. Scientifically, Atman is known as Spirit or Energy and Sansar or Nature is known as Matter. This energy and matter are incontrovertible as per Physics too. Quantum Physics explains the whole theory with proper scientific proofs. This all was already mentioned in Vedas thousand years back by our honourable and spiritual Saints, Sages and Gurus. That is the main reason why all Quantum Physicists are inclined towards Hinduism in the later phases of there research. Hinduism and Sanatan Dharma has all the solutions to every problem in the world.
More meaningfully, the body is a prison and the soul is held in it as a prisoner for the sins of the physical self and past deeds. At the time of death, the physical body and the gross mind returns to the elements of the earth. But the part of the subtle body survives, it accompanies the soul to the next world and hence the next life. Depending upon the nature and records of their past deeds (Karma), and the extent of subtle bodies they develop, the soul (Atman) either ascend to the ancestral heaven (pitra lok) or descend into the hell. Hiding the indwelling spirit in its core, the subtle being stays in these worlds until the fruits of its good or bad actions are fully exhausted. Having squared off the karma and learned new lessons, it then returns to the earth to take another birth and repeat the process. This is the whole technical process of Reincarnation.
Reincarnation is the concept which explain that a body dies, but the soul remains forever. Body is temporary but Soul is eternal. A body acts as a Car and the driver of this car is Soul. When we die, the identity changes to Body instead of our name because something was there who was driving this body, which is no more. A person who was having his own thought when he was alive is no more when he dies, so where does all the thoughts gone? It was soul which acts as a driving force and now moving on in the Sansar as per the Karma record. Reincarnation is the basic and unique concept of Hinduism which keeps the Cycle of Birth and Death as intact. No other religions accepts the concept of Reincarnation but Science itself has proved now that Energy and Matter are inter-controvertible. Renowned scientist Albert Einstein has proved this in his Classic Theory of Relativity about the formula of Energy and Matter conversion. E=mc
We also know that there are a lot of stories where a person remembers everything from his past life, it is an evident proof to prove the Reincarnation.