The foolish demand self-existence. They imagine they will take another body at death. Yet there is no "self" that transmigrates from life to life. Reincarnation refers to reformation, not transmigration. It is all about change, the body of which is karma.
Reincarnation refers to the continued life of the imputed I, the ego, which is a false thought, or ignorance. Just as one lights a candle from the flame of another, so are false thoughts passed from person to person. These thoughts are false because they believe they experience loss and gain, life and death. It is because these thoughts believe in life and death, they are said to be caught in the cycle of repeated birth and death. This false grasping mind experiences the inevitable changes of life badly, which prevents it from breaking out of the vicious circle of ignorance.
Some things appear and are true, like the sun. Other things appear and are not true, like happiness and suffering. Yet they still appear, and we have names for them. Reincarnation is such a name.
Buddhas are supposed to be beyond life and death, yet, in a sense, even they reincarnate, for they do not escape change, and change is ultimately birth and death. However, Buddhas do not experience the pain of change, which is birth and death.