Weyer studied as a young man with Heinrich Cornelius Agrippa, which matters for reading him: he came out of a learned magical culture, not out of hostility to it, and he never doubted that spirits act in the world. His medical training gave him something else, an eye for what a body and a mind under strain will produce.
The argument he built in 1563 runs like this. Demons exist and deceive. Old, poor, isolated women, often unwell and often in the grip of melancholy, are precisely the people a deceiver would find easiest to work on. When such a woman says she flew to a gathering and killed a neighbour's cow, the sane reading is that she believes it and it did not happen. She has neither the power nor the capacity. Punishing her is punishing an illness.
He wrote from cases. Weyer described patients he had attended, including people convinced of things that could not be so, and the descriptions are close, particular and unsentimental. He also insisted on a distinction his critics found intolerable: the learned man who sits down with a grimoire to summon a spirit acts knowingly and deserves what he gets, while the woman who confesses under questioning to a crime nobody could commit deserves a physician.
For a court, that was the usable part. It did not require a magistrate to abandon his theology, only to notice who was in front of him.