Christian monasticism is born in the Egyptian desert, basically with legends like eventual saint Anthony moving out of town into a cave to battle demons that tempt him in the form of beautiful women (and other things). The idea is to divorce yourself from "the world" and create a sort of paradise zone on Earth.

But two things happened with Anthony. First, thanks to the fact that he didn't just start walking into the desert and keep walking until he died, but in fact he stayed rather close to a town, he acquired a reputation as a holy man and prophet. People visited him to seek spiritual advice.

Second, as the legend goes, other people were inspired by him to take up similar forms of life. Some of them would end up living near each other, and eventually band together into communities. Of course the reality would not have depended nearly so heavily on one man's initiative, but the legend was useful for publicity purposes in late antique Egypt (and would continue to be through the Middle Ages--Lives of the Desert Fathers is a bestseller of the 15th century).

But this idea of forsaking "the world" to focus on God is the heart of medieval Christian teaching. Hermits were seen as especially pious because they accomplished this goal physically as well as mentally-spiritually. This physical separation could entail removal to the woods (the desert of medieval Europe), like the hermits that the knights of the Arthurian sagas are always visiting for advice. Or it could mean permanent enclosure in a single stone room attached to an urban church, with a window where the anchoress could watch the Eucharist being consecrated, like Julian of Norwich. (And if you lived in the early medieval Byzantine Empire, it could involve living on a platform on top of a pole. To my knowledge, the medieval West never developed a stylite tradition.)

And like with the Anthony of legend, medieval hermits' reputation for heightened piety and closeness with God led people to seek them out as spiritual counselors. So Margery Kempe visits Julian for advice, for example, or townspeople flock to witness Margareta contracta of Magdeburg perform feats of physical asceticism.

The foundation of a hermit's life, throughout the Middle Ages, was prayer. Physical asceticism--flagellation, periods of starvation, not bathing for long periods of time (no, really, this was a sign of holiness), exposure to rough weather--would have played a part. Yes, hermits tended to spend part of their time giving advice to people seeking guidance.

This is less applicable to Benedict's time, but by the later Middle Ages, even hermits needed access to the Eucharist--whether they were priests who could perform Mass for themselves and so just needed to acquire the physical bread and wine, or women/unordained men who needed a priest to perform the consecration. So the "desert" remained more an ideal than a reality.

So where does that leave us with Benedict and his Regula?

Around Benedict's time, there are a lot of people interested in some sort of super-pious life (or parents interested in the financial possibilities of placing children therein), and a variety of ideas of what that life looked like. Benedict's Rule laid out one version. The various religious reforms of Charlemagne's day did a lot to propagate the ROB as the vision of western monasticism, but it's important to remember that how individual houses lived out their monastic life continued to differ. OP, you're interested in Hildegard--she even wrote a commentary interpreting the ROB for a community of monks who wrote to her seeking guidance! Classic Hildegard, she stressed moderation in following its precepts.

So basically, hermits are the physical embodiments of the medieval Christian ideal: forsaking "the world" to focus on God. This material display of holiness writ bodily made people striving towards that ideal spiritually while embroiled in the day-to-day trenches of work and fighting see hermits as channels to heaven. So they were sought out for advice and marvel.

Monastic communities develop because the spiritual path of being a hermit, alone, is hard; they become entrenched in medieval European society as crucial social, political, and economic players as well as for their religious functions of praying for the souls of their patrons and the surrounding community.

How did his Rule gain such influence?

Benedict of Nursia himself was given a great deal of his fame from Gregory the Great's Dialogues, and his rule was used and spread based in large part on this fame. But it was only one of about a half-dozen popular in the seventh and eighth centuries, and many modern scholars are justifiably skeptical as to what "following a rule" meant in this period. There is a great deal of evidence that a monastery might use a rule as a starting point, but add so many local variations it would have been unrecognizable. Thus, we are hesitant to talk about "Benedictine monasticism" before the Carolingian reforms of the ninth century and the efforts of Benedict of Aniane (lots of people, including medievals, confuse these two Benedicts) to standardize and promulgate Benedict's Rule. And the idea of a "Benedictine Order" as a unit to which monks and foundations would self-identify as belonging really does not get going until the foundation of Cluny and the spread of its practices in the tenth century, and maybe not until the creation of other orders in the twelfth.

And why was hermitage such a prevalent cultural pattern? How did people know what Benedict was up to if he was a hermit?

The hermit is generally a popular form of the holy man, and the scriptural precedence in Christianity is in figures like John the Baptist. It was so popular that many early ascetics were, like Romanus and Benedict in turn, beset by others wanting to learn how to become hermits or obtain other benefits from association with the individual's sanctity. But living alone in the wilderness kind of sucks, and most realized this wasn't a course everyone could follow. Monasticism as an institution emerged as a sort of collective hermitage, and Benedict's rule is actually designed as a moderate sort of devotional life which most could practice. If they were successful, they might then themselves become hermits. In short, be careful about taking a modern idea of monasticism and pushing it back in time as a structure that existed wherever someone claimed to be a monk.

There's not really a work of recent scholarship where all of this is compiled yet, but whenever Alison Beach and Isabelle Cochelin get their Cambridge History of Medieval Western Monasticism out, I would go read that.