The people of the Renaissance did not live in "the Renaissance." The name for the period was settled by 19th-century historians, more than two hundred years after it ended.
The idea is older than the label. In 1550 the Italian painter and biographer Giorgio Vasari published his Lives of the great Italian artists. He argued that classical art had decayed after the fall of Rome and that recent Italian artists had brought it back. He called this a rinascita, Italian for "rebirth." In the 14th century, Petrarch had already contrasted his own age with the darker centuries after Rome fell.
The word that stuck was French. Old French renaissance meant "rebirth," usually in a spiritual sense. It came from renastre, "grow anew," which goes back to Latin renasci: re-, "again," plus nasci, "to be born." Before borrowing it, English spoke of the "revival of learning," a phrase Etymonline dates to 1785.
English took the word from the French phrase renaissance des lettres, the revival of letters. Etymonline records it in English in 1840, and as an adjective by 1842.
Then two historians turned it into an era. In 1855 the French historian Jules Michelet published a volume titled Renaissance. In 1860 the Swiss historian Jacob Burckhardt published The Civilization of the Renaissance in Italy. Burckhardt's book made the word widely known. He presented the period as a turning point toward the modern world, marked by a secular society and the importance of individual creators.
Historians have argued with that picture ever since. Many point to continuities with the Middle Ages, or to how religious the period actually was. The name itself carries a judgment: to call something a rebirth, you have to believe something had died.
The general sense, a revival of anything long in decline, appears by 1855. It now covers everything from the Harlem Renaissance of the 1920s to a Beyoncé album. Matthew Arnold tried to replace the French word with a more English Renascence from 1868. It never displaced the French form.