The programmatic claim, set out in the lectures published in 1873 as Introduction to the Science of Religion, is that religion can be studied the way language is, comparatively and historically, and that a religion known on its own is not really known at all. His formulation, adapted from Goethe on languages, is that whoever knows one knows none. It is the founding argument of the discipline and it is still the answer to anyone who asks why a scholar of one tradition should read about another.
His own comparative theory has not survived. He held that myth arises from a disease of language: that words for natural phenomena, especially the sun and the dawn, lost their transparency as languages changed, and that the resulting confusion was mistaken for narrative about persons. On that reading a large proportion of Indo-European myth turns out to be about the sun. The theory was mocked in his own lifetime, most memorably by a parody demonstrating that Muller himself was a solar myth, and the fieldwork generation abandoned it entirely.
His position was awkward in other ways. He was passed over in 1860 for the Boden professorship of Sanskrit, which went to Monier Monier-Williams, in a contested election in which his German birth and his liberal theology were both used against him. And he wrote as a believing if unorthodox Christian who expected comparison to show religion to be universal rather than to dissolve it, which satisfied neither the orthodox nor the secularists of his day.