Article CXVIII: Transmigration

In Chapter II, Verse 22, Krishna analogizes the transmigration of the soul with the changing of clothes. Clothes can be understood as the physical body where the person underneath represents the Self within. In this analogy, death represents the removal of old clothes, and birth represents the donning of new clothes. When the clothes are changed, the person underneath continues to be unaffected. Similarly, only the physical body ends with death while the Self continues unchanged. Reincarnation implies that the Self enters a new body at birth, which is analogous to how a person dons new clothes. Consideration and appreciation of reincarnation allows one to become less attached to one's own physical form as well as the physical form of others.

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Bhagavad Gita: Chapter II -- Translated by Swami Chinmayananda

19. He who takes the Self to be the slayer and he who thinks He is slain, neither of these knows. He slays not, nor in He slain.
20. He is not born, nor does He ever die; after having been, He again ceases not to be; Unborn, Eternal, Changeless and Ancient, He is not killed when the body is killed.
21. Whosoever knows Him to be Indestructible, Eternal, Unborn, and Inexhaustible, how can that man slay, O Partha, or cause others to be slain?
22. Just as a man casts off his worn out clothes and puts on new ones, so also the embodied-Self casts off its worn out bodies and enters others which are new.