"a remedy which is very effective, almost magical;" see silver (adj.) + bullet (n.). The belief in the magical power of silver weapons to conquer foes goes back at least to ancient Greece (as in Delphic Oracle's advice to Philip of Macedon). In Britain, silver bullets as a superstitious countercharm figure in the fictitious Popish Plot (1678).
'Cause champed silver kills stone-dead
Such as are musket-proof 'gainst lead.
[Thomas Ward, from "England's Reformation," 1710]
English folklore beliefs recorded from early 19c. held that a witch could be wounded or revealed (if transformed) only by a wound from a silver bullet. Similar fancies are reported in folk-tales from Ireland and Iceland. The belief in the killing efficacy of silver bullets was transferred to vampires by 1816.