The text reads: ".. in July 2016. Theoretical physicist Ali Kaya is one of them. He was arrested three months after the failed coup and held for more than a year before his trial took place. On 20 December ..." — that sounds like December 2016 to me, not 2017.

No, it reads: "He was arrested three months after the failed coup and held for more than a year before his trial took place. On 20 December, a court declared him guilty of being a member of a terrorist organization and sentenced him to six years of imprisonment

The coup was in July 2016. He was arrested after 3 months (October 2016) and after waiting for more than a year in prison, the sentence was declared. This makes it December 2017. Anyway, that doesn't even matter. 6 minus 1 year is still 5 years, and the articles reads as he's already out. What am I missing?

The article states:

"[They] released him early owing to the time he had already served in prison while awaiting trial."

It does. But he served 1 year while awaiting trial. So 6 - 1 = 5. He should still be serving almost 5 years now?

There is something called conditional release, where you get to leave prison early. Maybe he benefited from that. Basically, now that all prisons exceeded maximum capacity, if you commit a crime that requires up to 5 years of punishment you don't have to go to jail. You can directly benefit from a conditional release. For example if you steal something, and you don't have any previous criminal record, you walk free due to overcrowding.

In Turkey, prison sentences seem long on the paper, in real life you always get released a lot early. Especially now that prisons are hugely overcrowded.