I study ancient history and the Gobleki Tepe site has come up in a few of my courses, albeit only briefly. I mention this because I am far from an expert on this subject, but I can provide some basic academic information about the site, although I must immediately concede that there does not seem to be any kind of definitive answer to your question: we simply don’t know.

Gobleki Tepe was likely one of the first large-scale religious sites. In that period most people lived hunter/gatherer lifestyles in relatively small groups. The site itself was so large that it would have required a good number of these groups to work together to finish it in any decent amount of time (even then it probably cost several years). What is even more remarkable is that there is evidence that the buildings were regularly filled with gravel and sand: generally it is estimated about once every generation or so. After filling the old buildings people would rebuild the site on top of the old one. This would suggest that part of the site’s function lies in its ritualistic rebuilding, although we have no definitive answer as to why they kept filling and rebuilding it.

Reconstruction efforts haven proven difficult and there are various different ideas about what the original buildings looked like. For example, it’s unclear if they had roofs or not, wether buildings were sunken into the ground or not, or wether pillars were free standing or supporting a roof.

The pillars show reliefs of existing animals, but there are none of what would likely have been the main source of meat of the builders: gazelles.

The site is so large that the workforce required to build the place would not have been able to sustain itself through their usual hunter/gatherer lifestyle, not least because of the fact that it lies in a relatively isolated location, some distance away from an easily accessible source of water. The fact that a lot of food was required has led some to believe that the site may have been one of, if not thé, location where agriculture first took hold. The builders removed weeds allowing more room for certain edible wild grasses which could provide them with food.
Some have also theorized that people were tempted to build the complex by the group’s leaders, who threw feasts to lure people in. Why the leaders wanted to build the complex is of course up for debate. Perhaps they wanted to extend their influence, or leave something lasting, but that’s all purely conjecture on my part.

None of this really answers your question. Our best guess is that the site had some significant ritualistic purpose (why else invest so much time and effort every few decades) but what exactly is unclear. Its isolated location seems to support this claim, as ritualistic sites can often be found in relatively isolated locations.

I do not have a lot of literature at my disposal at the moment, so I had to draw most of this from my classnotes, but I can provide you with at least two articles that go a bit more in depth and also provide some of the same basics that I have mentioned above:

• Dietrich, O. et al 2012, “The role of cult and feasting in the emergence of Neolithic communities. New evidence from Göbleki Tepe, south-eastern Turkey”.
• Schmidt, K. 1998, “Beyond Daily Bread: Evidence of Early Neolithic Ritual from Göbleki Tepe”.

The most recent paper that I've seen substantially adds to understanding of the site. In 2017, the journal Science published "Modified human crania from Göbekli Tepe provide evidence for a new form of Neolithic skull cult" by the German archaeologists Gresky, Haelm and Clare (the German Archaeological Institute have been the principal investigators at this site for decades).

Science only very rarely publishes archaeology or anthropology-- when they do, its something important, and this is pretty dramatic (although that's dramatic to me, an historian-- and this is archaeology and anthropology, not history).

Key point in the abstract:

We focus on three partially preserved human skulls, all of which carry artificial modifications of a type so far unknown from contemporaneous sites and the ethnographic record. As such, modified skull fragments from Göbekli Tepe could indicate a new, previously undocumented variation of skull cult in the Early Neolithic of Anatolia and the Levant.

The three modified skulls from Göbekli Tepe represent an entirely new category of find, which testifies to the interaction of the living with the dead at this important Early Neolithic ritual center. These skulls, most likely removed from the postcranium in the frame of secondary burial rites, attest to the special postmortem treatment of certain individuals at Göbekli Tepe. Special status of the individuals could have been emphasized through the application of decorative elements to the crania, which were then displayed (also suspended) at designated points around the site. At present, it is unknown whether these treatments were performed in the frame of ritual activities in the monumental buildings or were brought to the ritual center from settlement sites within its catchment.

There are other sites where heads/skulls are removed from the deceased, notably Ain Ghazal (a site in the present day city of Amman, Jordan). But of course -- the fact that this was done at this site-- that doesn't mean that's what the site was "for"; stone lasts a very long time, and a site might have been built for one purpose or religion, then appropriated for another.

There are grooves in the skulls, and holes drilled in them-- eg it appears that these body parts were processed, post mortem, for some display or other effect, this wouldn't likely be for burial. Just who these people were -- notables, defeated enemies, objects of sacrifice-- is unknown. But we can look around the world at other cultures that process human bodies in similar ways to get some ideas of what _might_ have been happening. Again, it is all anthropology and archaeology, zero history here.

The full article from Science is here:

https://advances.sciencemag.org/content/3/6/e1700564

also see:

Kuijt, Ian, et al. “Neolithic Skull Removal: Enemies, Ancestors, and Memory [with Comments].” Paléorient, vol. 35, no. 1, 2009, pp. 117–127.

Bonogofsky, Michelle. “Cranial Modeling and Neolithic Bone Modification at 'Ain Ghazal : New Interpretations.” Paléorient, vol. 27, no. 2, 2001, pp. 141–146.