In the Upaniṣadic literature we learn that man is on a journey through samsara (the cycle/chakra of birth and death) having acquired countless forms because he does not realize the ātman which is not subject to samsara (the transmigrant is the jiva—never ātman). Man is like the rider of a chariot whereas the various forms he takes up are like the constructed chariot. Samsara continues as long as man cannot distinguish the rider/ātman from the chariot/body, the latter being a pseudo-ātman.
In light of this, it is hard to imagine the common explanation of Buddhism which many believe runs counter the Upaniṣadic view. But I shall take a go at it.
According to this view, man is on a journey through samsara having been reborn many times because he believes in ātman. But this ātman is an imaginary, false belief with no corresponding reality. When he looks at the five constituents (skandhas) which comprise his body which are: material shape, feeling, perception, volitional formations and consciousness, he finds no ātman (S., anātman). Finding no such thing this is believed to be Buddhism's great truth. So when the body dies, that's it. In other words, when the experient which is the five constituents goes out like a flame or fire that is the realization of nirvana. No more experient, no more suffering. But then for anyone familiar with the discourses of the Buddha, Buddhism is not saying this.
Permit me to have a go at explaining Buddhism.
Man is on a journey through samsara (the cycle of repeated births and deaths) because out of desire, he clings to the five constituents (skandhas) which comprise his body, believing that these constituents are his ātman. But they are not the ātman because they suffer, and by clinging to them we likewise suffer. We are not to regard the five constituents to be our ātman.
“But monks, an instructed disciple of the pure ones...regards material shape as: ‘This is not mine, this am I not, this is not my ātman;’ he regards feeling as: ‘This is not mine, this I am not, this is not my ātman;’ he regards perception as: ‘This is not mine, this I am not, this is not my ātman;’ he regards the habitual tendencies as: ‘This is not mine, this I am not, this is not my ātman;’ he regards consciousness as: ‘This is not mine, this I am not, this is not my ātman.’ And also he regards whatever is seen, heard, sensed, cognized, reached, looked for, pondered by the mind as:’ This is not mine, this I am not, this is not my ātman’” (Majjhima-Nikaya, i. 136).
When we can clearly distinguish our ātman from the five constituents this is nirvana. Said another way, when we can clearly distinguish the unconditioned from the conditioned, this is nirvana since it is detached from all conditionality (clinging to conditionality/the five constituents is necessary for samsara to occur). Of profound interest, the Upaniṣadic reading is not far from Buddhism in the Pali Nikayas in which we learn that we are not anyone of the five constituents. My ātman, i.e., what I really am, is not these constituents which comprise my corporeal body. The culprit is not the ātman but man's clinging to the five constituents.