They were primarily bi-lingual
Note: unattributed links are to general knowledge found on Wikipedia and primarily for the historical background.
Alexander the Great made his conquests during the early 4th c. BC, at which time Hellenization first occurred in the area (which in part actively sought the teaching of the Greek language).
The Celtic invasion of Galatia came approximately 50 years later, in the late 3rd c. BC, but the Celts would fall to Roman domination about 90 years later in the late 2nd c. BC, and "by the 2nd century AD the Galatians had become assimilated (Hellenization) into the Hellenistic civilization of Anatolia."
Regarding the Romans themselves, Michael Paravati notes:
the general population was diglossic, whereby there are two languages
in a given society which are used for separate functions, as opposed
to just simply bilingual, which would suggest that the two languages
are used interchangeably without as much discretion.
The second language being Greek, as Rome, though conquering Greece, was still itself Hellenized by the Greek influence of culture. The Greek language was used in different ways by both the upper and lower classes of Rome.1
Regarding the Galatian situation, Mark Janse's "Aspects of Bilingualism in the History of the Greek Language" specifically focuses part of its discussion around the central Asia area (where Galatia is).2 Part 3, the section on Cappadocian Greek in central Asia, offers a number of insightful points related to your question. Cappadocia was next to Galatia, and Janse mentions Galatia in some of the discussion (the discussion is largely about all of central Asia, but his focus of the paper in part is on how Cappadocian Greek came to differ from other Greek).
He notes Hellenization was accelerated before, and then reinforced by, the "Roman annexation (17 AD)" (348). The process was slower in the "rural areas," which maintained indigenous languages (349). This implies two things—(1) the rural areas became more bilingual (retaining indigenous languages), which also implies the cities moved away from those languages to Greek. Both city and countryside had varying levels of bilingual use, because regarding the regions around Galatia, "most of them are known to be bilingual in the ﬁrst century AD" (349).
Janse also quotes Vryonis that "'at an earlier period the Celts had been similarly Hellenized'" with respect to speaking the Greek language (350). The earlier period, here, however, is not wholly conclusive, since Janse/Vryonis are speaking of the 4th c. AD. But given the previously noted info about Hellenization of the region in early 1st c. AD, that is probably the "earlier" time being referred to. So 1c. AD is the point of becoming bilingual, but the Galatian region remains so for some time, as native languages are still spoken as late as 5th c. AD (351).
Being bi-lingual, most would have followed along with one reading aloud in Greek (even if they may have not been literate themselves to read). This is because a part of the Hellenization process also involved the culture of Greece, including its religion, so religious language would also be part of what was adopted by societies using Greek. In short, there was probably not much of a language gap to fill with the letter being written in a language known to have become "the common lingua franca of much of the Mediterranean region and the Middle East."
1 Michael Paravati, "Greek and Latin bilingualism beyond the upper class in the ancient Roman Principate," The Collegiate Journal of Anthropology 1 (Oct 2011).
2 Mark Janse, "Aspects of Bilingualism in the History of the Greek Language," in Bilingualism in Ancient Society: Language Contact and the Written Text (Jan 2002).