A popular book positing that theory is "How the Irish saved civilization" by Thomas Cahill, but to the best of my knowledge that idea isn't widely accepted as the main avenue of preservation of knowledge as the Muslim caliphates generally had many more preserved copies of Greek texts and it was during times of increased contact with Muslims that these texts reentered circulation. If you read anything by Bertrand Russell (though many people consider his thinking outdated) he put forth the view that Muslim science the middle ages was primarily notable for preserving western knowledge rather than innovating on it's own.

However, monks did indeed copy texts, though since this was done by hand it was time consuming and few copies were made. I'm not qualified to answer whether they did this in response to "barbarian" incursions or simply in response to the natural decay of the physical texts themselves.

The answer to your question is "sort of true" and "not true" depending upon your choice in the timing of key transmission moments, with the later situation being dependent upon the earlier.

Sort of true if you're talking about the immediate post-western Roman era, with the decline of Greek scholarship in the latin west. A few key elites who sort of straddled as religious/secular aristocrats (although I wouldn't call them monks in the high medieval sense) like Boethius, Cassiodorus, and Isidore were important in the translation of Greek works into Latin, thus rendering them accessible to the Western European audience as the Roman Empire was giving way to barbarian kingdoms.

Not true if you're talking about the "monks" at the scriptoriums of the Carolingian renaissance during the early middle ages. This being because it was not, "Christian monks saving western culture from marauding barbarians", but "Christian monks copying classical culture to further the cause of their 'barbarian' masters."

Except at this point, the barbarians weren't "barbarians." They were the new Romans (and the new Israel, and the new Trojans, and a whole host of other identities the Franks like to wrap up into their legitimization project).

And as stated previously, the latter dependent upon the former. If the first generation of post-roman copyists hadn't existed, there'd be less for the later carolingian copyists to work with.

But keep in mind, even the first generation of copyists were not copying to "preserve knowledge from barbarians" in expectation of a collapse of civilization, they were copying to render knowledge more easily accessible to their reading population (i.e. Latin). It's just that the continued and ultimately permanent breaking of the mediterranean as the early middle ages unfolded, there was less access to both greek knowledge and greek language learning, except in that which was already written in latin.

Which of course, was rectified by the high middle ages, when mediterranean communications began to re-emerge, via the crusades, and heavier trade with the eastern and islamic world, you see Greek literature (and of course greek language learning) filter back into Europe.

But with all that said of course, some knowledge was lost over time, but that happens with all evolving cultures and civilizations. It's just you have to keep in mind the idea of a re-emergent "western culture" coming out of the dark ages has as much to do with the Renaissance portrayal of itself as a re-emergence, rather than any sort of actual re-discovery.

Which means, the tale of the monks saving western culture is a useful story to tell people who are interested in portraying our current culture as a brave survivor of a time where knowledge might have been lost.

The reality of course (if it even exists, but this is a philosophical question), is dependent upon your perspective and the narrative you ultimately want to tell. If you're trying to write an anti-renaissance narrative, you would simply claim that that period in Europe merely "appropriated" a different culture's ideas and claimed themselves as its inheritors.

Civilisation was never threatened - people generally liked the nice things they had and they didn't want to lose them. The material standard of living declined, but it was gradual and it was only significant in Western Europe, where the Western Roman Empire had ruled ineffectively for decades before its end. The 'barbarians' never rampaged across the empire either, as most groups were fairly Romanised and Christian. Plenty of cities were sacked and libraries burnt, but those things weren't unusual in times of war and some leaders had tried to limit their impact. The famous Sack of Rome in 410 for instance was, as far as sacks go, rather civilised, since people taking refuge in churches weren't harmed and the goal was more about taking booty rather than destroying the city. More generally, 'barbarian' rulers took care to gain the support of the local Roman population once they settled down, because these people manned their governments and paid them taxes!

Lastly, monks (and non-monastic scribes) had been copying stuff for centuries by that point. Knowledge never entirely disappeared, since plenty of monasteries in the East weren't threatened by war. There were problems in the West, but that was because war and the decentralisation of power prevented the diffusion of knowledge/texts easily between different institutions. Without monasteries, we would have very few texts from Antiquity, but they weren't copying manuscripts to save civilisation - they were already doing that anyway, as manuscripts are fragile and needed to be copied frequently for future use.