Plato, the Greek philosopher, lived from 428 to 348. One of his principle concepts was that the soul is the essence of a person, being, that which decides how we behave. He considered this essence as an incorporeal, eternal occupant of our respective individual mortal physical presences. As bodies die the soul is continually reborn in subsequent bodies.
In Plato's time, human understanding was limited to what could be perceived by the naked eye, and much of philosophy was preoccupied with imagining the structure and meaning of the universe, without the help of higher mathematics, empirical research, or any enhancements to perception (such as telescopes or microscopes).
People in Plato's time, undoubtedly understood that sexual intercourse, for instance, leads to pregnancy, and that human reproduction proceeded by this process. However, the manner by which this actually occurs was unknown to them, since they had no conception of microscopic phenomena, the cell as the building block of animal or plant animus, or how sexual fertilization takes place. This enormous hole in their knowledge led them to posit all kinds of esoteric theories about how children are "created" in the womb, and how children inherit characteristics from their parents.
These concepts were mystical in origin, and were often included as a part of the metaphysics or theological systems of the time. Since Plato lacked the crucial information about how sexual reproduction actually works, in detail, he was obliged to supply a replacement description, for which there was no verification or proof whatsoever. Identity, for Plato, could only be "passed" through the medium of a soul, an incorporeal essence which entered the body at birth, and passed out at death. This soul was eternal, and passed through time, from body to body--in effect, the process of reincarnation.
It wasn't until the work of Gregor Mendel [1822-1884], and Darwin [1809-1882], that science finally began to understand how inheritance actually works, how the descent of species occurs at the level of sexual fertilization, creating a unique, hybrid individual from the DNA of separate parents, combined at conception. The mechanistic nature of this process would probably have astonished Plato, since, for him, the supernatural quality of the mysteries of life tended to be regarded as only possible through some kind of divine intervention, through means beyond human comprehension or description.
The idea that human evolution could proceed according to elementary combinations of strands of living DNA, according to laws of probability and chance, without apparent control from any external influence, would certainly have made Plato nervous. Human form and character the result of the gradual adaptation through the "accidental" occurrence of opportunistic mutations? Heresy!
But Plato's theory, as fundamentally naive as it now seems to us, was probably about as logical an explanation for genetic variation and "identity" as one could have made. Without the knowledge we now possess about genetics and natural selection, our hands would really be tied.
Things which we can't explain, because we don't have the information and data to analyze them, often lead us to make absurd speculations about how things work. The inception of the concept of the "soul" dates back well before the revolutions of knowledge which have explained vitality, aging, death, inheritance, and descent and selection. We continue to use the word "soul" to describe a host of qualities and conditions, because it's a handy term, a place-holder really, for things which we either can't explain, or for which we already have commonplace rational explanations, but which we'd prefer not to think of.
When people die, there is "nothing left" of their essence, their livelihood. The only part of them which may be said to extend beyond death is that itzy bit of DNA they passed on through their fortuitous encounter with the opposite sex. Unless, of course, you count their thoughts, or writings, or other artifacts (such as art). I have no descendants, my only child having died without issue. I am the end of my "line." For some people, this is a depressing thought; but I'm not bothered by it. Each individual lives and dies, and we all die eventually. Nothing that any of us "carries" in our memories or sensibilities will outlive us. It's all temporary.
But we also give some thought to posterity. We would rather the human race move towards something better, a perfected state of life. That's unselfish, and noble, and worthy.