What is Reincarnation
Reincarnation is an idea as old as civilisation itself. Most ancient tribal cultures have some form of theory that points to reincarnation of sorts. The basic premise of reincarnation is that when you die your life ends and you return to the earth, but in time you will reform into a different body.
This idea makes an assumption that a human is a construct of different elements, in the same way that a soup is a collection of vegetables and water. To notice these different elements in a person you can look at a dead body and ask yourself: what is the difference between the body being dead and the body being alive?
This may lead you to the conclusion that the dead body is not animated, it does not move or react. The human is divided into two elements: the driver and the car. The body is the car which does not do anything on its own and the driver is the animating principle which brings the car to life. This animating principle is often called consciousness, spirit or life force.
Body and Soup
The human is a collection of elements, two of which we will categorize as body and spirit, like how the soup is made of vegetables and water. If you put the soup over a hot stove for a long time the water begins to boil and evaporate, and if you keep this up eventually the water evaporates entirely leaving only the vegetables. If you didn't know any better, you would say that the water disappeared.
This is not what has happened: the water did not disappear, it merely changed its form. It went from being a liquid to a vapour. It left behind the heavy vegetables and became part of the invisible air, probably to form a cloud that will one day reconvert back to water and fall as rain.
Matter and Energy
I won't propose the existence of a soul, there is no evidence, but I will maintain the body and soul duality for the rest of this piece. This is because it coincides with another prominent duality in modern science: the duality between matter and energy.
Matter is our bodies, and the vegetables in the soup, but the water is also matter. The opposite, energy, would be the heat from the stove. This heat animates the matter and causes it to change through time, it causes the water to expire and vanish and it causes the vegetables to wither and burn. Energy is the animating principle which animates matter, and your consciousness, spirit or whatever you wish to call it would be the animating principle of your body.
The Conservation Of Energy
There is a very well established scientific phenomenon called the Conservation of Energy. This phenomenon is a study of systems. A system is a group of isolated things that act together. For example the soup pot would be the boundary of the soup, so everything inside of the soup pot would be part of the soup system. The entire universe is considered as one big system, like a giant cosmic soup.
Conservation of Energy describes how inside of a system, energy can never disappear, it can only change form. Much like how the water can change from liquid to vapour to solid but it can never escape the earth's system. The energy is always present, in the same amount, it just jumps around changing forms. Energy never dies.
Matter is An Illusion
Albert Einstein proposed in the early 20th century that matter and energy are different forms of the same thing, known through his formula E=mc2. This means that within any system there is only energy in various forms. Matter is actually energy, moving at a very slow speed, and when you animate it sufficiently it converts into energy. So once the water has boiled off the soup, if you keep the heat on very high, eventually the vegetables will burst into flame: releasing some of their stored energy as heat.
The Reincarnation of Energy
Your body is a system, like the earth, and like the soup. You put energy into daily in the form of food and the energy leaves you in the form of breath and exercise. Your body is part of a much greater system called the universe, within this universe exists everything we will ever know, and energy never is destroyed in this massive system, it exists as long as the universe exists.
There seems to be an animating energy that leaves the body at the moment of death, known as consciousness, the soul or the force (ala Star Wars). When this animating energy leaves your body it does not die, it changes form and reintegrates into the cosmic system. What happens to your memories and personality cannot be known, but the energy that once gave you life may someday change form and rain down on the earth to give life to a new body.
It may have been doing this since the beginning of time.