Herodotus actually did not mention any notable gardens in Babylon in Histories, and he is a notable absence in our sources.
Ctesias first described the gardens in his Persica, a book that was lost to us, but Diodorus Siculus quoted some of the sections in his Bibliotheca Historica, including a very detailed description of the Hanging Gardens and its architectural structures. The relevant sections are in Bibliotheca Historica, 2.10.
Ctesias was a physician who served in the court of Artaxerxes II; he likely witnessed certain splendid terraced gardens himself - since the architectural description was insanely detailed, down to the size of the supporting galleries and how trees could grow in the terraces - and then decided to attribute the gardens to Semiramis.