I can talk a bit about how this tradition got started, but I know very little about how it died out unfortunately. The stylites, and other ascetics like them (including my favourite, the dendrites, who lived in trees!), were powerful and respected because they stood outside of the confines of the secular world. They lived in the spiritual 'desert' on the fringes of society, granting them, in the minds of contemporaries, a privileged access to God. This meant not only operating outside of secular government and the local ecclesiastical structure, but also meant enduring physical and mental privation in order to be closer to God. Various forms of monasticism/asceticism had been popular for a long time already by the time Symeon the Stylite showed up in the fifth century, so he was only the latest in a long line of ascetic 'super-stars' in the region, but he was probably the first one to live on a pillar permanently. In this case, Symeon stood on the border-line between the Christian and 'pagan' worlds, as his pillar was a clear personal challenge to a nearby pagan pilgrimage site and invoked memories of past religious rituals concerning holy stones nearby. There is no doubt that he was a devout Christian, but he was still heir to the complex multicultural and multifaith legacy of the Roman Empire, making his popularity perhaps less unlikely than it sounds, since he did not act too far outside of the mainstream and also demonstrated significant continuity with the larger context.

Holy men like Symeon were not only influential because of their overt display of piety however, since they also served a social role. This is helpfully illustrated by this classic article from Peter Brown, 'The Rise and Function of the Holy Man in Late Antiquity' (1971). As outsiders from beyond the established social hierarchy, these ascetics acted as neutral arbiters in an increasingly uncertain world. Symeon certainly did a lot and his hagiographies are filled with stories of him arbitrating disputes and healing people's ills through prayer. In short, it is perhaps useful to see a holy man of the Syrian desert as something solid that a local villager or townsman could depend on in their daily lives, no different from rituals or patterns of behaviour we all like to rely on when we feel a bit uncertain. It seems clear that the locals deeply admired Symeon and a following quickly gathered around his pillar, eventually culminating in him giving advice to an emperor and various important bishops. He acted as the social glue that allowed society to function smoothly and a man who was by all accounts incredibly committed in God: in an age of faith, why wouldn't he have a seemingly incredible amount of influence?

His disciple, Daniel, took it even further, as he built a pillar in Constantinople and soon developed strong connections with the imperial court. As such, when the usurper Basiliscus came to power and sought to alter ecclesiastical policy, Daniel was able to decisively shame the emperor and force him to abandon his plans - the stylite was simply a ridiculously popular holy man in the capital with powerful allies, someone who could not be ignored by an usurper in a relatively weak position. This story only survives in a Life written after Daniel's death, a work that sought to promote his sanctity and his cult, so the story was probably more complicated than that, but I don't see any reason to doubt its fundamentals. This I think covers the gist of the functions of the stylite saints, but let me know if you have any more questions about this.

About Byzantine history, I think either Judith Herrin's Byzantium: The Surprising Life of a Medieval Empire (2008) or Averil Cameron's The Byzantines (2009) is your best bet. They both write very well for a general audience and they are genuinely some of the biggest names in Byzantine studies, so they are I think the two best overviews of Byzantine history. Peter Sarris' Empires of Faith: The Fall of Rome to the Rise of Islam, 500-700 (2011), Mark Whittow's The Making of Byzantium, 600-1025 (1996), and Michael Angold's The Byzantine Empire, 1025-1204: a Political History (1997) are meatier books by specialists and more geared towards students making their first forays into Byzantium, but they are still narrative histories and so should still be relatively accessible if you want more details :)