The exact mechanism of cellular development remains poorly understood, but we are now able to grow entire animals from a single cell, and it doesn't matter which cell is the starting point. The DNA is 'stuck' to the host cell, somehow. The now viable egg, turned zygote, is implanted into a surrogate mother who gives birth normally. The infant animal appears identical to the original at birth, but as environmental factors weigh in, differences begin to develop. This all seems to be a bit bland, hardly worth the hoopla that surrounds it, until one entertains a bit of speculation.
Animal cloning is a natural extension of the genetic engineering techniques now being developed.
Animals and plants can be exploited on a trait by trait basis given the ability to have a perfect supply. Our food will be tastier, our medicines will be cheaper and more plentiful, who knows, maybe the perfect wine grape will be developed. Well to start with, it seems anti-biological to clone oneself. Your own clone would have all of your trait assets, benefit from your mistakes and share your needs and desires. Your perfect competitor, considering its youth.