Chinese names are highly concentrated, both surnames and given names: today, 5% of the common family names account for 85% of the population and specific given names tend to be fairly numerous too. In one survey based on the 1982 census, the top 28 given names accounted for 30% of the people! Then add in the sheer number of homophones in Chinese. Furthermore, many times given names contained components (usually a specific character) that marked the individual child in relation to others: birth order and clan generational order being the most common. Given names thus had a tinge of being formulaic about them at times. So, for several reasons, there was pressure to find monikers that mark one as truly individual.

In imperial eras, elite males could adopt a "style" name (zi 字 or biaozi 表字) when they came of age, often at the "capping ceremony". ("Style" is a confusing term and I agree with Endymion Wilkinson that the field should standardize the translation to "courtesy name" because it was a respectful form of address intended for use outside the family.) Girls from elite families could get a courtesy name when they got married too, or at age 15 sui if they weren't yet married. Often male courtesy names included a marker of age seniority with one of the polite terms for (high ranking) men. Sometimes they were matched to the original given name of the individual in some way (including puns).

Often the true "courtesy names" are hard to distinguish from men's "alternate names" (hao 號). (Some English sources, for example, use "style" to designate hao; sometimes it is hard to determine which is which in the historical sources too, however.) These alternate names could be self-selected: zihao 自號 or given by others zenghao 贈號.

Men adopted such alternative names for lots of different reasons: to celebrate an achievement, to mark a life transition, to identify themselves with a particular place, or a new artistic direction. They allowed literati to adopt personas or craft their literary-artistic identity in particular ways and thus were most popular among writers, painters, calligraphers. They generally didn't completely stop using their surnames or given names when they did so. Up until the Song, men usually limited themselves to one hao. From the Ming onwards, some men chose many more -- a few, particularly painters, seem to have 50-100 different hao. (There are dictionaries for looking up zi and hao of individuals.) The practice had stopped by mid-20th century.

Source for the above: Endymion Wilkonson, Chinese History: A New Manual, New ed., Harvard University Asia Center, 2012, pp. 112-147.

As for the question of why the practice was abandoned, I don't know of any definitive source or study on that, but there very well may be (especially in Chinese). Some of the sources listed by Wilkinson may address this issue, though I would think he would mention it if they had. I suspect Euro-American practices of one person, one name (through the May Fourth Movement and the New Culture Movement) was one influence. I wonder, however, if the expansion of the modern surveillance state (beginning in the late 1920s) is another important cause: a population that can change its name willy-nilly makes regulation, tabulation, tracking, etc., much more challenging for the state bureaucracy. Marriage practices, for example, went through a similar process of bringing unruly social practice increasingly under the regulatory gaze of the state during the 1920s-50s. Source: Susan Glosser, Chinese Visions of Family and State, 1915-1953, University of California Press, 2003, passim.

EDITS: minor word changes for grammar.