This question has countless possible answers.
Much of the ancient world believed in rebirth, the Romans, the Greeks, the Celts, the Africans, the Asians, the Sumarians, the Hebrews, the Norse, the Aryans (India), the various Pagans, etc. etc.
Today, much of Judaism accepts reincarnation (human form to human form) as do many Christian mystics. Buddhists and Hindus and others typically believe in transmigration (any life form to any life form), and so on.
For many Jews, every "Jewish soul" that has or ever will live was present at Sinai for the giving of the Torah. At death, some people go to Gan Eden, a sort of heaven, some reincarnate and a few may be destroyed if they are deemed by God to be unworthy, but this is rare.
A similar view is held by Gnostic and Mystic Christian sects and there is evidence the belief was quite common among Christians before Emperor Constantine established the Universal (Catholic) Church and created its dogmas that most Christians still accept today. There are hints to it
in both the Hebrew and Christian Bibles.
For most Buddhists, existence is an illusion based upon false identification. While one remains in ignorance one returns to the material universes time and time again until finally awakening to the Oneness of Nirvana. At that time, the being realizes its eternal existence as No-Self,
a No-Thing (as opposed to no self and nothing which have opposites). The being simply Is. For Mahayana Buddhists (the majority), a vow is taken to forgo this liberation until all beings attain it.
For Hindus who embrace the Advaita systems the situation is similar. They transmigrate until realizing the Neti-Neti, the Not this Not That, the I am That (tat tvam asi). As with Buddhists, they don't cease to exist, they attain to unity with all that is and forgo (rather than lose) independent existence.
For most Hindus, the material conception may be thought of as a limited schoolhouse (some would say prison, but school is closer I think). While we are materially clothed we are learning and realizing so many things. In this process we go through countless life forms and experiences.
The human form of life is precious and special because as humans we have reason and the ability to evaluate our situation. From this form of life we have the opportunity to escape karma and graduate from the wheel of rebirth, to return to the our eternal positions in the spiritual
worlds through the various yoga systems. This view is closest to my own.
So as you can appreciate, there are many different views here. I'll be happy to discuss any of them with you in more detail if you'd like.
As for why people believe these things, there are many reasons as well. In nature we behold life beginning in the spring with a seed, maturing, fading and seeming to die in the fall, only to return the next spring. We see loved ones born, age and die only to be replaced by the next
generation. We see this cycle throughout nature. Seeing it, we assume rebirth to be the way of things and we assume that we too, like the grasses of the field, will return in due time.
Also many of us have memories from our past lives that we are convinced are true. I began believing in rebirth even though everything in my strict Baptist upbringing denied it, because in light of my dreams nothing else made sense. (http://allfaith.com/Mystory/dream.html).
Scriptures like the Vedas teach this directly. The Bhagavad Gita says:
Gita 2:12:
Never was there a time when I did not exist, nor you, nor
all these kings, nor in the future shall any of us cease to be.
2:13: As the embodied soul continuously passes, in this body, from boyhood to youth to old age, the soul also passes into another body at death. A sober person is not bewildered by such a change" (My translation:
http://www.allfaith.com/Religions/Hinduism/gita2.html)
This deeply-held belief has always existed and doubtless always will. It satisfies our longings, gives us hope and faith to carry on during adversities, and minimizes the fear of inevitable death. Death is not the end, it is merely a door through which all will pass. Every thinking person wants some idea of what comes next, and reincarnation, in its various forms, meets this with logic, heart-truth, remembrances and expectations.
As for heaven, most rebirth concepts include a heaven and, usually, a series of hells. These are usually temporary in duration however. In Hinduism for instance they are called the Naraka planes.
I believe we never stop growing, never stop developing, even though I fully believe we will leave these material planes at some point and continue our schooling in realms we can not currently even conceive.
Hope this helps,
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