Jumbo was a real elephant, and he became so famous that his name turned into an adjective. What nobody can say for sure is where his name came from.
He was born in what is now Sudan around 1860 and captured as a baby. He passed through an animal dealer to the Jardin des Plantes in Paris. On June 26, 1865, he arrived at the London Zoo. There he grew into the zoo's star, looked after by his keeper, Matthew Scott. His shoulder height has been estimated at about 10 feet 7 inches, though his showman owner later claimed more.
In 1882 the zoo sold him to the American showman P.T. Barnum for £2,000. The British public was furious. Wikipedia reports that 100,000 schoolchildren wrote to Queen Victoria asking her to stop the sale. It went ahead anyway.
The word followed the elephant. Etymonline dates jumbo, "very large, unusually large for its type," to 1882, the year of the sale. By 1886 it was being used for sizes of cigars. Jumbo himself died on September 15, 1885, struck by a freight train in St. Thomas, Ontario. The adjective outlived him and later gave us the jumbo jet, attested by 1964.
The name is the mystery. English already had a slang word jumbo, "a clumsy or unwieldy fellow," recorded in 1823 in John Bee's slang dictionary. Where that slang came from is uncertain. Etymonline notes a suggested West African source, comparing Kongo nzamba, "elephant." The OED suggests a link to mumbo jumbo.
Other guesses point to Swahili, either jambo, "hello," or jumbe, "chief." These are popular but unproven. The etymology blog Mashed Radish surveys the theories and concludes that nobody knows exactly who named Jumbo or what the name meant.
What is certain is the second half of the story. An elephant named Jumbo became a celebrity. And the celebrity became the word for anything extra large.