I have finally managed, I believe, the put my beliefs about reincarnation into words. I've tried before, and it was close, but I could never quite get out exactly what I believed. There were still lingering questions and the like attached to those attempts to describe my beliefs. With that said, this is the gist of what I believe about reincarnation, souls, etc.
Everything is made of energy, from to plants to rocks to animals to people. Energy never goes away, but it can be transformed. I believe that everything in the known and unknown universe is made of energy, with some things carrying more than others (ex. a rat would have more of this energy than an ant.)
Everything, living and inanimate, are like tiny pieces of the Universal energy. Their separate components also have bits of this energy (think of the composition of minerals present in stones: each component has it's own energy, then those components together create the overall energy of the stone.)
The soul is also energy, but more specifically it is the energy that animates things and gives them personalities. This part of the energy is shaped by the person or thing it inhabits, and traces of that person/thing remain after the soul leaves the body. More on this later when I talk about past-lives.
When something dies, it goes back to the earth and the energy that gave it life is reunited with the Universal energy. The soil takes in the physical and the energy from decomposition can yield new life, richer soil, etc. Think about the Lion King and the "circle of life." The body decays, which lets new things grow and then be consumed by something that will in turn die, decay, be consumed, etc.
See, this energy does not disappear, it is transformed. It begins as something living, dies and decays, something grows where it decayed, something else consumes the plant and is nourished by it, and the cycle continues.
To explain how the world has more people now than ever before, while still believing in reincarnation, think about how species become extinct, and so forth. Their energy doesn't disappear. So the energy of dinosaurs, as an example, would still exist, it just transformed, became embodied by something else.
As for past lives, as I said before, the imprint of a person lingers in the bit of Universal energy they were made up of. When the energies of the deceased transfer into someone else, because it's all the same energy in different forms, it's possible, in my opinion, for someone to remember pieces of the past.