A FabLab actually doesn’t “function” on its own… It’s even a bit difficult to precisely define what a FabLab is. It largely depends on what the users of the FabLab make of it.
I think that the Wikipeadia page on Hackerspace decribes a bit better than the original page what a FabLab is: “a community-operated workspace where people with common interests […] can meet, socialize and collaborate”. While Hackerspaces typically focus a bit more on computers and software, a FabLab typically offers more machining capabilities like CNC mills, lathes, laser cutters, etc. and is less software and computing centric.
Anyway, if you want to make something in such an open-space, you will mostly have to do that on your own. Which doesn’t mean that you won’t find help - not at all! For most of the machines in such a space, you will find someone who is already more experienced using it than you will probably be - and they will be happy to help you. But they won’t take over any of the tasks or jobs, that you as an inventor or maker need to do to make your project successful.
What the FabLab in Lübeck offers is an excelent set of tools and a lot of enthusiasts. But actually no staff that will do your work for you. You will have to waste your own time and resources and only pay a very small usage-fee for the machines.
And finally: the FabLab equipment is not only available to the Fachhochschule Students and Staff but to everyone who is willing to collaborate in the open workshop.