How did right handedness win over left handedness in numbers?
Is it only a coincidence that there are more right handed people than left handed ones? Or, has some effect in nature explicitly made right handedness dominant?
First of all, asymmetries in apparently symmetric creatures (of which most are) are actually quite typical. However, in most cases of hand dominance, there is no population-wide hand dominance. In other words, the population is split 50/50 between left and right handers. In humans, however, a lack of hand dominance is often associated with cognitive deficits.
In humans, population-wise, we are right-hand dominant (9:1). There are exceptions to a genetic influence, such as strict schools disciplining children into using their right hand, or an indirect consequence of developmental brain trauma or genetic diseases, but for the most part the right-hand dominance is thought to be a genetic outcome based on the emergent effects of several interacting genes. That is, there's no particular reason why nature "chose" right over left, just that having a dominant hand allows for more specialization with respect to tool-using which could possibly have given humans with this mutation a competitive edge over their ambidextrous competitors.
A single-gene theory (Corballis, 2009) proposes that there is a dextral and a chance allele of a particular gene and that people born with the dextral allele are likely to be right-handed while people born with the chance allele have equal chance of being left or right-handed.
Francks, C; DeLisi, LE; Fisher, SE; Laval, SH; Rue, JE; Stein, JF; Monaco, AP (2003). "Confirmatory evidence for linkage of relative hand skill to 2p12-q11". American journal of human genetics 72 (2): 499–502. (PDF)