You can download the article here
[1308.4526] Formalization, Mechanization and Automation of G\"odel's Proof of God's Existence
It's a bit more coherent, and provides some of the necessary context, as well as discussing the actual assertion being tested.
A mathematical proof does not prove that something exists; it proves that the logical propositions are consistent in their format and conclusion. In math, there are lots of things that are "proved" that do not actually exist anywhere except in mathematics. Now, if you're some version of a platonist, that's fine, but if you reject all versions of platonism, it's pretty meaningless.
Godel showed that in any logical system, it can be complete (in which case it cannot be consistent--there will be paradoxes) or it can be consistent (in which case it cannot be complete). The only way to prove some assumptions (things that we assume to be true because if they aren't, the logical system will fail) in any logical system is by finding a larger logical system.
So, basically what the authors have shown is that the proposed proof of God is logically consistent within a certain logical system.
I'll also note that at the link above, the last revision of this article was in 2013, and as far as I can tell, it has not been updated since nor accepted for peer-reviewed publication. I'm not sure why the university press office decided to issue a story, which was then picked up by Der Speigel and the Express. Maybe the authors were trying to stir up some interest, so it might get picked up by a peer-reviewed journal...it happens...