And it's actually more plausible than you might think. The logic goes something like this:
As more people moved to towns more people started wearing underwear and underwear made of better quality materials produced by professional weavers rather than homespun garments.
And we're being very general here - we know from marginalia in manuscripts and from other sources that the sight of peasants with their wedding tackle dangling between their tunic hemline was a source of ribaldry, as was the sight of fine ladies falling of horses and inadvertently showing their bums. So think chemises and shifts rather than knickers. Knickers only really become common when people start wearing trousers and care about skidmarks.
At the same time as towns grew there was an increased demand for records and documents as life became more complex. Co-incident with increased urbanisation papermaking became common in western Europe, and to make paper you need rags, preferably good quality rags with long fibre lengths. And source for these was the worn out linen shifts. And because there were more people in towns in meant that there was enough good material available for rag collectors to operate profitably and sell the rags to the papermaker.
And suddenly what initially seemed a loony idea is suddenly sensible - it's not that increased wearing of underwear resulted in increased literacy, it's that the increased use of finer materials for undergarments provided a ready source of raw material for papermakers when these garments were discarded.
Compare this with the Roman period, where a fair proportion of the population was literate, but lacking paper wrote on wood slats, pottery fragments as well as papyrus.