The Upanishads believe in a changeless core of the self called as the "Atman." Untouched by the variations of time and circumstance, the Atman was entrapped in the world of "samsara" (the cycle of death and rebirth). Unlike western ideals of reincarnation, life in this world means suffering.
An individual is trapped in the samsaric cycle because of the law of karma. In its simplest form, this law operates impersonally like a natural law, ensuring that every good or bad deed eventually returns to the individual in the form of reward or punishment commensurate with the original deed.
Coming back in another lifetime also allows karmic forces to reward or punish one through the circumstances. Therefore, an individual who was generous in one lifetime may be reborn as a wealthy person in the next incarnation.