These clues from the Galápagos led Darwin (immediately? long afterward? here the mythic story is vague) to conclude that Earth's living diversity has arisen by an organic process of descent with modification—evolution, as it's now known—and that natural selection is the mechanism. He wrote a book called The Origin of Species and persuaded everyone, except the Anglican Church establishment, that it was so.
Well, yes and no. This cartoonish account of the Beagle voyage and its consequences contains a fair bit of truth, but it also confuses, distorts, and omits much. For instance, the finches weren't as illuminating as the diversity of the islands' mockingbirds, at least not initially, and Darwin couldn't make sense of them until a bird expert back in England helped. The Galápagos stopover was a brief anomaly near the end of an expedition devoted mostly to surveying the South American coastline. Darwin hadn't signed on to the Beagle as its official naturalist; he was a 22-year-old Cambridge graduate pointed rather indifferently toward a career as a country clergyman, invited on the voyage as a dining companion for the captain, a mercurial young aristocrat named Robert Fitzroy. Darwin did assume the role of naturalist, and think of himself that way, as time went on. But his theory developed slowly, secretively, and The Origin of Species (full title: On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life) didn't appear until 1859. Many scientists, along with some Victorian clergymen, resisted its evidence and arguments for decades afterward. The reality of evolution became widely accepted during Darwin's lifetime, but his particular theory—with natural selection as prime cause—didn't triumph until about 1940, after it had been successfully integrated with genetics.
Apart from those clarifications, the most interesting point missed by the simplified tale is this: Darwin's first real clue toward evolution came not in the Galápagos but three years before, on a blustery beach along the north coast of Argentina. And it didn't take the form of a bird's beak. It wasn't even a living creature. It was a trove of fossils. Never mind the notion of Darwin's finches. For a fresh view of the Beagle voyage, start with Darwin's armadillos and giant sloths.