This is a vague term, with no exact definition, but which has become associated with one particular organization.
Technically, it refers to a “room for reading,” but is often used to mean a reference library, which is a repository of books that may not be checked out – they have to be read on-premises, and therefore the facilities have a room specifically for reading.
However, the Christian Science Church has popularized the term by providing what they call “reading rooms” in many cities, which are really just libraries because they loan, give away, and even sell books as well as provided space to read them.
So, a library, or even a bookstore, with space for reading can be called a “reading room.”
Why I Looked It Up
In a book about the history of the Koch family, the patriarch, Fred, wanted to create a series of “anti-communist reading rooms” across the country where people could read and contemplate the works of great anti-communist writers and thinkers.