I have shortened a fascinating and enlightening article, A Review of Sexual Selection and Human Evolution, by G. F. Miller. This is the core of the information that I found the most compelling.
Darwin (1859, 1871) realized that his theory of natural selection through differential survival could not explain extravagant male traits such as the peacock's tail, because such traits actually decrease survival ability. Rather, he reasoned that in a sexually-reproducing species, any heritable traits that help in competing for sexual mates will tend to spread through the species, even if they compromise survival somewhat. This process of sexual selection may favor, for example, better sensory and motor abilities for finding mates, gifts and ornaments to attract them, weapons and bluffs for repelling same-sex competitors, endurance for lasting through the breeding season, and genitals and gametes that maximize fertilization rates...
R. A. Fisher (1915, 1930) was one of the few biologists to take sexual selection seriously. He viewed mate preferences as legitimate biological traits subject to heritable variation, and this insight led him to postulate a process he called runaway sexual selection.. In runaway, an evolutionary positive-feedback loop gets established between female preferences for certain male traits, and the male traits themselves. Given a nudge in the right direction (e.g. an initial bias in female preferences), Fisher's model could account for the wildly exaggerated male traits seen in many species, such as the peacock's plumage...
Even chance fluctuations in mate preferences, combined with a strange kind of evolutionary positive-feedback loop, could produce quite extreme mate preferences and quite exaggerated courtship traits...
"Females that prefer to mate with long-tailed males will mate with such males more often than females that prefer short-tailed males. Following mating and genetic recombination, the genes for long-tail preference and the genes for the long tail itself will become correlated: an individual carrying a gene for long tails will tend to carry a gene for the corresponding preference."(Kirkpatrick, 1987, pp. 74-75).
But why bother with sexual selection? What's wrong with the traditional story that natural selection just generally favored intelligence, learning, tool-making, and culture? The problem is that the evolution of big brains is so rare, so recent, so capricious, and seemingly so unrelated to the demands of habitat or econiche (Miller, 1993). Brain size in our lineage has tripled over the last two million years, reflecting the evolution of unprecedented mental and behavioral capacities. Over three million years ago, our ancestors were already successful, social, fairly bipedal, tool-making hunter-gatherers on the African savanna — and they had brains only slightly larger than the chimpanzee's. Then, two million years ago, for no apparent reason, brain size started growing exponentially in our lineage but not in other closely-related hominid species who shared the same habitat, such as Paranthropus boisei and robustus. Encephalization then stopped about 100, 000 years ago, again for no apparent reason, long before the Neolithic revolution in technology and art 40, 000 years ago. Extreme encephalization also happened in some species of cetaceans (dolphins and whales) and proboscids (elephants) living in quite different environments, but has not occurred in other primates living in quite similar environments (e.g. baboons, chimpanzees, Paranthropus hominids).
The speed, uniqueness, and capriciousness of this encephalization process has prompted many theorists to accept that human mental evolution must have been driven by some sort of positive-feedback process that is sensitive to initial conditions. There have been two traditional contenders. In the runaway social competition model (Byrne & Whiten, 1988; Humphrey, 1976; Whiten, 1991; also see Miller, in press), hominids got smarter to predict and manipulate each others' behavior, leading to a social-intelligence arms race between mind-reading and deception. In the runaway gene-culture co-evolution model, hominids got smarter to learn and use material culture (e.g. tools and survival techniques), which was itself evolving (Durham, 1991; Lumsden & Wilson, 1982; Wills, 1993). Yet these theories overlook the clearest and best-established case of positive-feedback evolution in nature: runaway sexual selection. The runaway process is a good fit to the human evolution data because it begins and ends unpredictably, without much relation to the external environment, but it is extremely powerful and directional once underway (Miller, 1993; Miller & Todd, 1993).
If the brain evolved through runaway sexual selection, what were the relevant traits and preferences? Two uniquely elaborated aspects of the human brain are its creativity (Boden, 1991, 1994; D. Campbell, 1960; Freyd, 1994) and its neophilia, or love of novelty (Zuckerman, 1984). Perhaps creativity itself became a trait subject to sexual selection by neophilia as a mate preference. More technically, mental capacities for generating `protean' (adaptively unpredictable) courtship displays may have been subject to `neophilic' mate preferences in both sexes (on proteanism see Driver & Humphries, 1988).
The gradual evolution of language was especially important, because it allowed hominids to display complex ideas and images to one another using an increasingly complex, structured, open-ended, combinatorial system (Pinker, 1994). Language gave potential mates a unique window into each other's minds, so allowed much more direct sexual selection on the mind itself. Also, language permits gossip, which can transform mate choice from an individual decision to a social decision that integrates information from family and friends. With language and gossip, courtship displays need not be observed directly; they need only be witnessed by someone who can talk later to potential mates. The feedback loop between sexual selection, language complexity, and mental complexity was probably the mainspring of human mental evolution.