You must get to level 20 and obtain a chocobo license the normal way, just like the update notice says.
From what I understand, once you have a Chocobo license, the update makes it possible to ride a Chocobo in two new situations.
1) You switch jobs and level that job to 15. You can now ride a chocobo between level 15 and 20. Previously, you would have had to switch jobs back to a job that is over 20, or raise your new job to 20 or higher.
2) Presumably, if you were to level down due to deaths after getting your Chocobo license, you would still be able to ride them as long as you are still above level 15.
I have not tested #2, but I don't see why it wouldn't work.
I have made use of #1 several times now. Like the update notice says, you can only rent the Chocobo from any of the three main nations. (ie: Bastok, San d'Oria or Windurst). You cannot rent a chocobo in any of the other towns (Jeuno, Kazham, etc) or from the crags in La Theine, Tahrongi or Konschtat.
The rental guy will warn you that your riding time is being reduced, and that items and equipment that extend riding time will not work. I have never tested how long the chocobo lasts under these conditions, but it's long enough to make the journey from San d'Oria to Selbina without a problem.