I missed this question when it was first posted, but it deserves an answer (credit to /u/Casiphoner for flagging up the question). It in fact deserves a fuller answer than a non-specialist in medical history can give. I can, however, provide some medical-literary sources that illustrate some of the broader trends in late medieval thinking about sleep. The bigger picture here is that European medical theory, like natural science and mathematics, was heavily influenced from the twelfth century and onward by contact with the writings of Arabic scholars. A lot of this material built on the core principles espoused by Aristotle and Galen; that is to say, the four humours, the imbalance of which was thought to cause essentially every medical ailment. Moderate balance, conversely, was thought to promote health, and it is here that sleep seems to invite discussion.

One such text is the Secreta Secretorum or Secret of Secrets, a widely circulated Latin translation of the pseudoaristotelian Sirr al-Asrar. Several vernacular translations and adaptations were circulating in the later Middle Ages, but for ease I quote from John Lydgate's Secrees of Old Philosoffres, finished by Benedict Burgh after Lydgate's death in 1449. I quote from Steele's EETS edition (which is very old and probably flawed, but the best we have). There is a whole passage on healthful sleep (1261-95) that accords broadly with a holistic, humoral theory of medicine. I quote some choice lines (translation my own).

 

Sleep is noryce of digestioun

Yiff it be take in attemperaunce,

Yif slogardye yive Ony occasyoun,

Causith hevynesse slouthe or disturbaunce

Put a man Out of good gouernaunce,

Be war of wach kepe also the date,

To kepe a mesour of Etyng and drynkyng late.

 

Sleep promotes digestion, if taken in moderation. If given over to sluggishness, however, it causes dullness, [or sorrow] slothfulness and stress; it puts a person out of their right mind. Pay attention to the time, to moderate eating and drinking too late.

 

Notice that sleep is linked closely to eating - this continues throughout the passage, such as at line 1282, "Afftir mete be-war make no long sleep" (After food, take care not to sleep too long). It is also, in moderation, understood as a preventative and a moderator both of health and of behaviour - sleep correctly, and the rest will follow. The mechanisms of sleep aren't addressed, but there is a degree of self-justification involved: you need to sleep properly because, if you don't, these bad things will happen.

A similar sentiment is espoused in Lydgate's Dietary, compiled (probably in the second quarter of the fifteenth century) from a range of Galenic/Aristotelian sources, as well as more folk-like medical lore. The Dietary survives in 57 manuscripts, and was printed by Caxton, de Worde and Pynson; despite its total lack of narrative content, it was evidently one of the most popular poems of the fifteenth century. Here are some choice lines on sleep from MacCracken's edition (which is also somewhat old and dubious, but it's what we have):

 

For helthe of body...

Vpon thy sleep drynk neuyr of thi cuppe,

Glad toward bedde and at morwe, bothe too,

And vse nevir late for to suppe.

(25, 30-32)

 

And in especial flee meridian sleep.

(40)

 

All this processe concludith vpon tyme

...

The golden sleep braidyng vpon pryme

(89-91)

 

Lydgate's advice here follows the same thinking. Sleep at the proper times, for the proper length, in moderate relationship with food and drink, and it will promote good health and good behaviour. The poem catalogues a great many similar (slightly platitudinous!) pieces of advice about living moderately and well, and concludes with a gentle jibe about the more arcane medical remedies promoted by apothecaries.

But what about lack of sleep? Here I quote (from the Riverside edition) a purely literary source, Chaucer's Book of the Duchess (c.1370). Bear in mind that Chaucer borrows more than a little from Machaut here:

 

I have gret wonder, be this lyght,

How that I lyve, for day ne nyght

I may not slepe wel nygh noght;

I have so many an ydel thoght

Purely for defaute of slep

That, by my trouthe, I take no kep

Of nothing, how hyt cometh or gooth,

Ne me nys nothyng leef nor loth.

(1-8)

 

I am astonished (by this light!) that I still live, for neither by day nor night can I sleep even a little. I have so many idle thoughts because of my lack of sleep that, truly, I take heed of nothing; neither how it comes nor goes, nor is anything to me lovely or loathsome.

 

Chaucer is writing fiction rather than literary medical advice, of course (his sleeplessness becomes a pretext for a dream vision), but he draws heavily on Macrobius' Dream of Scipio, an important source of medieval dream theory - which is its own significant area connected to sleep theory. Again, sleep is understood as a regulatory function; without it, Chaucer's narrator's very perception of reality is thrown off-kilter. His cure is similarly holistic. All the things missing in the narrator's life - sleep, emotional engagement, social interaction, even the enjoyment of books! - are resolved together through the dream vision. It seems plausible that the moderate balance espoused in humoral medicine influenced Chaucer's thought here.

As I alluded to above, it is rarer for these sources to address the mechanistic explanation for sleep - that would be the realm of faculty psychology and, in all likelihood, rather more specialist, prosaic Latin treatises. Hopefully someone else might be able to give a fuller answer on those? These more literary and advisory texts, at a remove from the "hard science" (if we can think of it like that?) can seem a little circular in their logic. "You sleep because it is good for you" is perhaps not so different from "You sleep because you need to." But these works do, I think, give an interesting sense of the circulation of the basic principles of advanced medical theory outside of specialist works. It is worth questioning (although I do not have an absolute answer!) the degree to which a medieval non-specialist would have desired - or even been able - to access more dense, mechanistic explanations of sleep.