At first I wondered whether this was mostly due to his engaging personality and his man-of-the-people Ulster accent; but as I explored his performances I realized that it was based on the fact that he really was the best flute player of the era - and by some considerable margin.
By 'best' I mean quite simply that he made by far the richest and fullest sound of any flute player (ever?), and also that his phrasing was extremely fluid and lyrical - and these are the qualities most valued by the general, music loving public at large (leaving aside the more various views of expert critics).
In other words, people like Galway for the same reason they like a great singer such as Wunderlich, Sutherland or Pavarotti - because they make the loveliest sound and also 'phrase' the music in a way that brings-out its beauty above all else. So by 'great' I mean 'middlebrow great' in the same way that Austen, Dickens and Frost are regarded as great, and not in the way that Joseph Conrad, Henry James or TS Eliot are regarded as great.
Anyway, if you haven't yet listened to Galway, then perhaps you should. Here he is playing JS Bach 'chamber music' - the Trio Sonatas. Although great favourites of mine, these pieces can seem quite dry and intellectual, the way most performers play them - but not Galway/ Chung!