How did that happen? Immigration?
Not now they don’t. There was a Celtic language spoken there thousands of years ago, but there were Celtic languages spoken in many parts of Europe thousands of years ago. Most are now extinct.
Galations were what Celtic tribes were called by the Romans it is the same root as “Gauls”. Recall that there was a pan-European Celtic presence from Ireland to Asia Minor – the later of which which was after all the Celtic homleand, where they lived before crossing and intermixing with Europe, changing the face and culture of Europe as well as what it meant to be a Celt.
Here is a site discussing the first Celtic encounters with “Western Civilization” (read Greece)
One tribe of Celts fought their way across the Balkans and Anatolia in something like the 5th Century B.C. and ended up settling in the area referred to as Galatia, now in East Central Turkey. They continued speaking a Celtic language, overlain with whatever was the lingua franca of the time (Aramaic, Hellenistic Greek, etc.) for several hundred years. References in books on the Celts indicate that there were still Celtic-speakers there in the early Middle Ages. The language is now extinct, and the people are mixed with other Middle Eastern peoples.
Far as anyone can tell, the Celts expanded out of central Europe, not Asia Minor.
I think you may have just misread your cite and/or conflated them with the Indo-Europeans, who a few scholars, based on linguistic evidence, think may have originated in the vicinity of Anatolia ( it is still a minority view, however ).
Actually, as far as I know, Galatian is considered to be a Celtic language, of the same subgroup as Celtiberian, Gaulish, Lepontic, and Noric, and equally as extinct.