Which books are essential for learning about it, or that have thoroughly collected it?
The classic, translated work in by Italo Calvino, Italian Folktales. Calvino (1923-1985) was a writer and journalist, drawing from other collected works, but it is difficult to get beyond his work since it is the go-to edition in English language for Italian folktales.
The sixteenth-century two-volume work by Giovanni Francesco Straparola (ca. 1485-1558) is a classic in European folktale collecting, representing one of the earliest published collections of European folklore, in this case exhibiting what was available in Italy at the time.
The fourteenth-century work of Giovanni Boccaccio (1313-1375), The Decameron, is also noteworthy for how it draws on early Italian folklore, much in the way that Chaucer did during roughly the same period.
You can also consider the work of Giambattista Basile (1566-1632) whose writings drew on Italian folklore.