The sequence appears in the Liber Abaci as one exercise among hundreds, a puzzle about how many pairs of rabbits there will be after a year if every pair breeds on a fixed schedule. It was not new. Indian writers on prosody had generated the same numbers centuries earlier by counting the ways long and short syllables can fill a line of a given length, and Pingala, Virahanka and Hemachandra are all cited for it in the Sanskrit tradition. The name Fibonacci was not his either; it was applied to him by historians in the nineteenth century, and it is that later naming which attached the sequence to him.
Its career since has mostly been outside mathematics. The ratio between consecutive terms approaches the golden section, and on that thread hangs a large body of claiming about the proportions of the Parthenon, the great pyramid, medieval cathedrals and Renaissance painting. Careful measurement studies of those buildings have repeatedly failed to support the claims, and the arithmetic used to produce them turns out to be flexible enough to find the ratio almost anywhere. That is worth saying plainly, because the sequence does occur genuinely and interestingly in nature, in the spiral arrangement of leaves and seeds, and the real case is obscured rather than helped by the invented ones.
Sources Liber AbaciLeonardo of Pisa (Fibonacci) with Misconceptions about the Golden RatioGeorge Markowsky, The Golden Ratio: A Contrary ViewpointClement Falbo