The Listening Center was basically a massively scaled-up and networked language lab of the type that was prevalent on American (and international) university campuses during the twentieth century. Language labs, in turn, are a way to implement the audio-lingual method of language instruction. This method holds that the best way to learn a language is to hear it spoken and then repeat or respond. This is in preference to studying the written language and to talking about the language's grammar, vocabulary, etc.

The audio-lingual method could be pretty intensive in terms of instructor time and resources, so in the early twentieth century schools that used it began to implement systems to automate it, such as rooms with booths in which students could listen to recordings of the language being spoken. These became known as language labs by analogy to the laboratories for science experiments. Typically there would be a phrase spoken on the recording, then a pause for the student to respond, and then a recording of the correct response. These began with phonographs and progressed to tape recordings. The recordings were preferably made by native speakers so the student would learn to emulate their pronunciations, idioms, etc. Many language labs also included recording facilities so students could record their responses for later criticism or for evaluation; e.g., a test in a course heavily using the audio-lingual method might consist of going to the language lab and recording responses for the instructor to listen and grade.

Ohio State University's Listening Center adapted the language lab model to a gigantic post-WWII American campus with tens of thousands of students. By this time, the dominant paradigm was what you might call the self-service language lab: a student who had time to knock out an hour or so of audio-lingual drill would visit the language lab, check out the correct cassette, and sit at a booth. But Ohio State was a huge facility, serving a massive population of students with different schedules, so Pimsleur and his colleagues scaled up the system and distributed the booths across campus while finding and using new technology to avoid having to distribute the cassettes. Instead of there just being one language lab, there were many all across the campus, and starting in the mid-60s additional booths were installed at off-campus residences such as fraternity and sorority houses. (Not exactly partying all the time!) The cassettes were stored centrally and made available to students on-demand through a dial-in system that could handle thousands of simultaneous calls. So the student with time available went to one of the distributed booths, called the central system to specify what program was needed, and then completed the listen by listening and responding to the program. This required what was, for the time, highly advanced telephone technology. Because the Listening Center provided a method for providing audio content to students on-demand, it was also used by some departments to provide access to recorded lectures.

Interactivity, in the form of letting instructors listen in on student lessons and respond to them, appears to have been added in the 1970s. This capability was added to language labs around the world and became standard by the 1980s.

Progress through the materials offered by the Listening Center would have been part of a language course at Ohio State so there was probably an expectation that students would progress at a certain rate, but for details beyond that generalization you'd probably have to look at course syllabi.

This should also answer your question about payment: it was part of the course of instruction for enrolled students. My understanding is that the telephone system used by the Listening Center was closed; people couldn't just dial from anywhere but had to be in one of the booths, to which access was presumably restricted. (I mean, if you wanted to bluff your way onto a college campus to listen to intermediate German tapes, I guess you could have.)

In the middle of the twentieth century, audio-lingual learning was certainly considered a legitimate way to learn a language, even mainstream. It had been the U.S. military's preferred method of language instruction during WWII. Language labs were widely adopted by American and international universities. When Pimsleur was at Ohio State in the 1960s-70s, it was considered one of the premier American universities not just for language instruction but for also teaching students how to teach languages, whether they were undergraduates who hoped to become high-school language teachers or graduate students on the path to becoming professors. The Listening Center just turned the audio-lingual language lab to 11 or higher.

Audio-lingual learning's reputation as a teaching method within universities has declined somewhat and it is now generally encountered as a part of a multi-pronged approach. E.g., students will have a mixture of classroom instruction, written homework, discussion sessions with other students, and drills using the audio-lingual method. With a wired campus and ubiquitous digital devices, there is no need for the particular language lab infrastructure and such facilities have typically been committed to other uses.

I suspect the OP got interested in this topic because of the commercial Pimsleur method. This began as Pimsleur's effort to provide the audio-lingual method of instruction to a broader public that was not enrolled as students, initially by selling them the cassettes. As was typical for commercial distance learning, customers were being offered a part of the way similar content was taught in the university, though with the audio-lingual method it was the primary part. The basic idea of learning a language by listening to tapes and responding was not unique to Pimsleur. Pimsleur's own specific refinements to the audio-lingual method had to do with the specific ways in which new content is introduced and then reinforced. Pimsleur was not all that successful personally running the business so he licensed it to others. After Pimsleur's early death, the product probably became more associated with his name and legacy. This should not give the impression that he was a huckster or quack; he was a respected scholar and his methods were firmly within the academic mainstream.