Humans, like all vertebrates, belong in subregnum bilateria, a broad class of animals whose characteristic trait is having a bilaterally symmetric body plan at least in some of their life stages.
The common ancestor of all bilaterians was presumably something like a small marine worm. For a primitive animal living in water, an obvious advantage of bilateral symmetry is that it makes directed swimming easier. If the animal were completely asymmetric, it would have to continually exert active control over its heading to be able to swim straight. On the other hand, an even more symmetrical animal (such as one with 90° rotational symmetry) might have trouble controlling its vertical and lateral heading separately, which could be a problem in aquatic habitats where staying at a certain depth is often useful.
(Indeed, in many ways these reasons are the same as why pretty much all aircraft are bilaterally symmetric: in the absence of active steering, we want them to fly straight and level. That requires at least rough left–right symmetry and generally also some degree of top–down symmetry, although some breaking of the latter is usually needed both for landing and to account for the effects of gravity on flight dynamics. We could build completely asymmetric aircraft if we wanted, they just would be harder to fly.)
As for why this ancestral bilateral symmetry has survived so well throughout the course of evolution, that's presumably both because it's so deeply embedded in the genes that control our ontogeny, but also because the basic reasons why such symmetry is useful still remain, even though our size, shape, habitat and locomotion are very different from those of the first "urbilaterian". Even though we mostly move by walking instead of swimming, it's still useful for us to be able to walk straight without having to pay constant attention to it. Furthermore, once we've first learned to walk (or crawl) straight, it's useful that we can also learn to run, swim and jump (and ride a bike or drive a car) straight without having to always re-learn the exact amount of control needed to maintain a given heading with each of these modes of locomotion.
Ps. To answer your actual question, I'd guess, like Rory M, that the two halves of a human body probably won't weigh exactly the same, both due to the asymmetrical distribution of the internal organs and also due to uneven muscle development.
However, the difference is quite small compared to the total mass of the human body, so that the center of mass is presumably still quite close to the body's centerline. As I noted above, any significant deviation from that would cause issues with gait and balance. Although such issues can certainly be adapted to and overcome with active control — after all, even people who've lost a whole leg manage to get around one way or another — they're presumably still significant enough to be selected against over evolutionary timescales, which is why our body shape remains so nearly symmetrical.