Christian monasticism has a very long and varied history, dating back close to the origins of the church.

Some of the earliest known were the Coptic ascetics of the Middle East, sometimes called the Desert Fathers, a movement beginning around the 3rd century CE. These were the earliest philosophers of the Church, and they followed an informal doctrine of "standing where Christ stands". Essentially what this meant was understanding God and sin through examining the life of Christ. The early ascetics found that the best way to do this was to isolate and seclude themselves in the deserts, leaving themselves to focus on self-reflection and introspection without the influences and distractions of society. These early ascetics weren't wealthy or scholarly, they were largely commoners (peasants, slaves, even prostitutes).

Over the centuries communities of like-minded individuals began forming around them, and monastic life began to take a more structured form; communal living focused on spiritual growth and reflection, and this way of life spread throughout Christendom rather quickly.

Around the 5th century we get the first concerted effort to structure monastic life into what we know of it today from St. Benedict of Nursia. Benedict was an aristocratic Roman, but after completing his schooling he fled to the Italian mountains to live an ascetic life (rumored to be near the ruins of Nero's palace). He constructed his Rules (now the Rules of St. Benedict) based on the philosophies of other ascetic philosophers, namely Pachomius, Basil, and Cassian, and following his return to regular life, built his first monastery at Monte Cassino.

The Benedictine order, like the Desert Fathers, was a focus on living "Christlike", and was centered around personal sanctity through prayer, liturgy, service, and personal reflection. Benedict's Rules were originally tailored to men, but he left his monastery open to women and children, and his Rules were adapted to both. This was also the beginning of the practice of oblation, that is, parents committing their children to monastic life.

The monks and nuns of the Benedictine order focused on communal prayer, devotional reading, and common household work (including fieldwork, these were practically self-sufficient communities). The goal was seclusion and isolation from the temptations of life, particularly money, sex, and ambition. Priests were welcome to provide sacrament, but it was not required for monks to be ordained and consecrated in the priesthood.

[EDIT]

Apologies, some of this is really poorly worded as I wrote it before my standard coffee injection. I'm going through and changing some things to make them clearer as the caffeine fills my veins and bestows lucidity.

Sources and additional reading:

• The Desert Fathers: Sayings of the Early Christian Monks - ed. Benedicta Ward (Penguin Classics)

• The Faith of the Early Fathers vol. 1 - ed. William A. Jurgens (The Liturgical Press)

• Journey Back to Eden: My Life and Times Among the Desert Fathers - Fr. Mark Gruber (Orbis Books)

• The Roman Catholic Church - Edward Norman (University of California Press)

• Medieval Europe: A Short History - Judith Bennett, C. Warren Hollister (McGraw Hill)

Somewhat in addition to what /u/spilurum explained: from the perspective of the monk and the Catholic hierarchy the monastery served multiple functions.

• It was a lot easier to live in an enclosed community under a strict rule than solo in a deserted or wasteland area, and the number of places which would support an Eremitic lifestyle were fairly few and far between. The monastery allowed monks of less steely constitution to devote their lives to God.
• The seclusion that the desert fathers sought was constantly being intruded upon by people who sought their blessing, which somewhat diminished the value of seeking solitude unless one went to great lengths to lose themselves in the wilderness. The monastery allowed people to seperate themselves from worldly matters with rules rather than terrain.
• From the perspective of the Church, the desert fathers were at constant risk of lapsing into heresy without the benefit of guidance and discourse with the wider community of theologians, which presented a risk to the pilgrims that sought out the desert fathers as well. The monastery allowed the monks to be watched and guided to prevent them from straying from the path or misleading others.

As to why it was necessary at all: the Catholic Church has always anticipated a final reckoning. The Western Roman Empire being overrun by Gothic peoples convinced a lot of zealous individuals that the second coming would be sooner rather than later; and people obviously wanted their souls to be in good standing when the hour arrived. As the years passed the terror subsided, but every new crisis (the Viking era was a major one) caused a surge in religious devotion, and centennial and millennial panics happened on a regular basis because the idea that God likes round numbers has always been a popular one.

From the perspective of a medieval ruler, the monasteries were an almost perfect vassal: the charter might stipulate taxes and the provision of knight service just like any other vassal, but their much more ethically strict worldview tended to limit their potential for treachery. The presence of a large library, scriptorium, and the highly literate and motivated inmates meant that a monastery was almost invariably a well run enterprise. The same factor made the institution a convenient place to send children to be educated, to store and publish documents, and to assist in the management of secular estates. Lastly, the institution itself provided an outlet for younger sons and brothers that had little hope of inheriting land, and a way for an aging ruler to relinquish the reigns, retire, and make their peace with God without having their male relatives shortening the timescale.