The claims of many European university to have been established in a specific year in the eleventh or twelfth century are dubious at best. They did not suddenly start anew, with brand new buildings, teachers and curriculum. They evolved gradually from monastic schools, and when they say that they were established in a specific year, that is usually the date of the earliest extant document that someone was able to find in the archives, and it does not tell us much about what the "university" was like at that time. In Paris, for example, there were at least four schools in the central middle ages, and at least two of them were attached to benedictine monasteries and were not actually public schools at all. We have to get to the thirteenth century to have a school with a reliable founding date.

In 1224, the emperor Frederick II established a new "studium generale" (that is what universities were called at that time; I have an older reply on the difference between "studium" and "universitas) in Naples, in his Kingdom of Sicily. Unlike previous universities which evolved organically from religious institutions into schools of broad learning of arts, law and philosophy as well as theology, the university in Naples was the first one to be established at a definite time by royal charter. A renowned school of medicine was active in Salerno, but for other subjects both students and teachers were travelling to Bologna and to other centers of learning through Europe, which was both a "brain drain" issue for the Regno as well as a political issue, since Bologna was aligned with the papacy and not with the empire.

The charter is quite interesting in what it says about the importance of university education at that time: “We wish that in all parts of the Kingdom many will become wise and knowledgeable ... and will become useful to us, for the administration of Justice and of the laws which we urge everyone to obey.” The society of the central middle ages was evolving, and there was a growing bureaucracy, both in the secular and religious world, and there was a parallel growing need for people trained in canon law and in secular law.

Now, the typical university student of the middle ages was not a late teen leaving home for the first time. Chances are that they were adults already working in a field (typically law, theology or medicine) who needed further education that they could get from the masters teaching in Bologna, Paris or many other cities in Europe. The education and knowledge were important and very valuable, as the way to get into well-paying positions. The degree itself, not so much. Many students would complete the studies and not bother with getting the actual degree, something that was very expensive and only gave you the additional right to teach at any university.

Sources:

Medieval Sourcebook. “Frederick II: Licterae Generales, establishing the University of Naples, 1224” http://sourcebooks.fordham.edu/halsall/source/1224fred2-lictgen.asp.

Hyde, J.K. “Universities and Cities in Medieval Italy.” in Thomas Bender, ed. The university and the city: from medieval origins to the present. Oxford University Press, 1991.

Oldfield, Paul. “The kingdom of Sicily and the early university movement.” Viator, 40.2, 2009, 135-150.

Pedersen, Olaf. The First Universities. Cambridge University Press, Cambridge, 1997.

Rashdall, Hastings. The universities of Europe in the Middle Ages. Vol 2. Oxford: 1987,