Monasticism is frequently associated with "the contemplative life", as opposed to "the active life". While there are many different kinds of monastics, an important subset is those who are chiefly engaged in maintaining a simple, prayerful life in an enclosed community, and are therefore found much more towards the contemplative end of the spectrum.
The contemplative life was often characterized as being more "elevated" than the active: a pursuit that not all were capable of sustaining. Isidore of Seville writes:
The active life is the innocence of good works; the contemplative life is the regarding of higher things. The one is common among the many; but the other among the few. 1
It is a common observation that the contemplative life was most suited to those given to contemplation - for example, being capable of thinking about one thing for a long time. Further, those with attachments to business affairs, or who sought marriage, were advised to embrace the active life. Gregory the Great associates the contemplative life with those who are dissatisfied with common life, having "troubled souls" (in amaritudine animae) as they are "crucified to the world" (Galatians 6:14) and suspicious of earthly rewards. 2
The active and contemplative lives are frequently allegorized by the figures of Leah and Rachel (Genesis 29ff), or by Martha and Mary of Bethany (Luke 10). Augustine writes, and is followed by many others:
[Leah's fertility] represents all the useful result of a laborious life exposed to the common vicissitudes; a life which many avoid on account of its troublesome engagements, because, although they might be able to take the lead, they are bent on study, and devote all their powers to the quiet pursuit of knowledge, in love with the beauty of Rachel. 3
This characterization, very common in the medieval period, associates the contemplative life (and therefore monastic seclusion) with people who are attracted to quietness, concentration, and intellectual study; but not attracted to the to and fro of making their way in the world at large. So monastic institutions could indeed function as a haven for "misfits", assuming they were misfits who were acceptable to the institution. (One does not generally just turn up and say, "Hey everybody, I want to be a monk", job done. Of course this is enormously variable by time, place and social status, but as a rule, one has to have an extended trial period before being accepted as a full member of the community. Unruly people might also be expelled.) Indeed, this applies not just to monasticism but to becoming a member of the clergy, at least in the case of male misfits.
I don't want to give the impression the medieval monasteries were populated entirely by docile introverts. On the contrary, many monks were not cloistered (even in theory), and maintained colorful social relations.
1. Isidore of Seville. Sententiae book 3, chapter 15, De contemplatione et actione. From Migne Patrologia Latina vol. 83: "Activa vita innocentia est operum bonorum, contempliva speculatio supernorum; illa communis multorum est, ista vero paucorum."
2. Gregory the Great. Moralia in Iob book 5, chapters 1-5.
3. Augustine. Contra Faustum book 22, chapter 56. "in qua est omnis fructus laboriosae atque inter incerta tentationum periclitantis actionis: quam plerique bono ingenio praediti studioque flagrantes, quamvis idonei regendis populis esse possint, tamen vitant propter turbulentas occupationes, et in doctrinae otium toto pectore, tamquam speciosae Rachel feruntur amplexum."