It was not man’s immortal spirit he meant. Rather, Buddha was pointing to the temporary physical, passionate, personal and psychic parts of us.
Our deathless spirit uses such vehicles merely as instruments of expression in each new life.
Functionally linked those parts are temporary vehicles and are separated from each other at death. They are reduced to their primal elements, like the fuel of a fire, recycled and returned back to Universal Nature, their primal state.
The process is entirely natural, the recycling of renewable substances of evolution, of the temporal forces and materials required for an earthly body:
“Dust thou art,” states Genesis (3:19), “and unto dust thou shalt return”— referring to those perishable parts of man’s complex construction.
By contrast, as in dream, “the Spirit of man is free,” as the occult teaching of the Brihad Aranyaka Upanishad (13) declares, “and naught adheres to the Spirit.” Mme. Blavatsky agrees with the Upanishads, and explains in the Key to Theosophy (109) that “death ever comes to our spiritual selves as a deliverer and friend.” And for the average mortal “it will be a dream as vivid as life, and full of realistic bliss and visions.”
Even for the materialist, who, “notwithstanding his materialism, was not a bad man, the interval between the two lives will be like the unbroken and placid sleep of a child.”
“As the man at the moment of death has a retrospective insight into the life he has led, so, at the moment he is reborn on to earth. He has a prospective vision of the life which awaits him, and realizes all the causes that have led to it.
“He realizes them and sees futurity, because it is between [the bliss filled after-death dream state called] Devachan, and re-birth, that the Ego regains his full [spiritual mind] manasic consciousness, and re-becomes for a short time the god he was —before he first descended into matter and incarnated in flesh, in compliance with Karmic law.”