Paul threatens Shaddam IV and the Spacing Guild that he'll destroy the spice unless the Guild moves the armada of his enemies away from Dune and Shaddam abdicates the throne. Somehow they decide he can really do it. How was he going to do this?
from Book III (just after Chani wakes Paul from his water-of-life-induced coma):
Paul took a deep breath, said: “Mother, you must change a quantity of the Water for us. We need the catalyst. Chani, have a scout force sent out … to find a pre-spice mass. If we plant a quantity of the Water of Life above a pre-spice mass, do you know what will happen?”
Jessica weighed his words, suddenly saw through to his meaning. “Paul!” she gasped.
“The Water of Death,” he said. “It’d be a chain reaction.” He pointed to the floor. “Spreading death among the little makers [immature/larval sandworms], killing a vector of the life cycle that includes the spice and the makers [mature sandworms]. Arrakis will become a true desolation—without spice or maker.”
Chani put a hand to her mouth, shocked to numb silence by the blasphemy pouring from Paul’s lips.
“He who can destroy a thing has the real control of it,” Paul said. “We can destroy the spice.”
Incidentally, since everyone who lives on Arrakis is addicted to the spice, Paul would also be condemning the whole resident population to at best, serious sickness, at worst, death from spice withdrawal.