This text I am reading says that the University of Berlin had a ‘universitas litterarum‘. What does that mean?
Grammar and Meaning. ūniversitās is a singular noun in the nominative case, meaning whole, body/group (of people), association, corporation, or (in late medieval/early modern Latin) University; litterārum is the genitive plural of littera; in the plural it can be used to mean one or more documents, records, or, in the context of education, it often means written knowledge and literature.
Since you have the genitive plural modifying the singular, this means literally something like "a University of literature," "a society of letters," "a community of letters," "a community of literature," "a community of written knowledge," etc. I'll have a bit more to say on this in a moment, but keep in mind that when the University of Berlin was founded (1810), all of these shades of meaning would be actively understood by classically-trained scholars and the motto is pretty surely engaging with the interplay between some (for example, including both universitas in the sense of an institutional university, but also in the more organic sense of a community of people engaged in a common task, etc.)
Meaning and Background: ūniversitās (f., -tātis) is literally a whole, a body or group of people, an association, guild or corporation, or (in late medieval and early modern Latin) a University in the sense of the academic institution of higher learning. The last, most familiar use comes from the increasingly common use of expressions (in charters, official documents, etc.) like universitas magistrorum et scholarium (association/community of teachers and scholars) or universitas studii (association/community of study) to describe the institutions and legal privileges of emerging institutions of higher learning.
Before there were Universities in the modern sense, there were towns like Bologna or Oxford, which developed reputations as centers of learning, which attracted both students seeking teachers and teachers seeking students, both of them often coming from far away. Both teachers and students started developing associations to help protect their professional and political interests (often hard for out-of-towners to do in the Middle Ages), and eventually also used these groups to help structure classes and educational curriculum. The oldest official charters for what we call Colleges generally came about from official recognition of the teachers' professional associations (collēgia); the ancient charters of what we call Universities (in this sense) usually developed out of official recognition of the students' unions (ūniversitēs). (For more background, see for example 1911 Britannica: Universities.)
Etymologically the term is directly related to ūniversus (world, whole, universe) and both seem to be transparently derived from something like ūnus + versus, i.e., "that which is turned/changed into one."
Meaning and Background: litterārum is the genitive plural of littera. In the singular this is the word for "letter" (as in written signs or marks for sounds, like A, B, C...); it can also be used to mean a written word or handwriting. In the plural, it can mean a bunch of letters, but it can also mean a letter in the sense of written communication (like in the mail); or to mean a document or documents; or to mean a record or account-book; or to mean literature. The last should be understood in the broad sense -- not just as meaning literature in the sense of written artworks (like novels or poetry), but in the broad sense of written knowledge (that an educated or cultured person may/should be familiar with; in both classical and modern Latin it includes not only the arts but also humanities, philosophy, science, etc.; so think not just of "literature" in modern English but also "the literature," in the sense that scientists and scholars use the term to refer to the organized body of writing in their field).
Wilhelm von Humboldt, in particular, was committed to a particular model of Enlightenment education, which focused heavily on the idea that educational institutions should aim not only at instructing students in a fixed set of traditional precepts or inculcating professional skills, but should emphasize the cultivation of individual students' powers of reasoning by engaging them in a community of current research, independent inquiry, and direct, humanistic engagement with primary sources, which should integrate the study of the classics, humanities, and up-to-date natural science, under the protections of independent research and academic freedom. The motto ūniversitās litterārum should be understood as emphasizing the importance of the university as an organized community which is engaged in a common project, and the connection of the common project to humanistic engagement with literature or "the literature," i.e. (1) a university that studies literature; but also (2) an active and engaged community of learning and research.
The University of Berlin itself was established as a universitas litterarum, which means "community of letters, of humanistic learning transmitted through writing". The first word simultaneously means "community, corporate society" but also "entirety", suggesting that the place offered a holistic education, and is where the word 'university' comes from.