Instead of "letters", it's more accurate to think in terms of *sounds*, since the *i* in those Italian words represents a "y-sound" like the first sound in English *yank*.
Here is the theory accepted by most Romance historical linguists:
In Latin words beginning with *pl-*, *cl-* (pronounced [kl]), *fl-* and sometimes *bl-*, in some parts of Spain as well as Italy, the /l/ phoneme took on a "brighter" sound, like the *l* in *some* English-speakers' pronunciation of *million*. Phonetically, the place of articulation moved from the alveolar ridge to the hard *palate*, so the change is called "palatalization".
The new sound at first maintained its *lateral* quality, meaning it allowed the airstream to flow to one *side* of the tongue, the same as with the original "l-sound".
After that, the Spanish and Italian dialects parted ways.
In standard Spanish, the initial consonant was lost, leaving the palatal lateral sound at the beginning of the word. Spanish spelling uses a double *l* for that sound, so in Spanish we have (Latin > Spanish) *plenum* > *lleno*, *clavem* > *llave*, etc.
In Italian, the initial consonant was preserved, but the palatal lateral lost its lateral quality (if you want a name for this change, call it "delateralization"). The result was the "y-like" sound as in *pieno*, *chiave*, *fiore*, etc. (Compare the pronunciation by some English-speakers of *million* as "miyun".)
This scenario is supported by a transitional form that preserves both the initial consonant and the palatal lateral (the "bright" *l*), in [Ribagorçan](https://en.wikipedia.org/wiki/Ribagor%C3%A7an_dialect#Phonology), a dwindling dialect of the Ribagorza region of northern Spain, transitional between Castilian and Catalan. When Spanish spelling is used to represent this dialect, the palatal lateral sound appears as double *l*: *plleno*, *fllama*, etc. Some spelling errors in 12th-century manuscripts suggest that the palatalization was happening by that time in Spain, but it's hard to pinpoint a date when this process began, because it was probably during a period of the early Middle Ages when early Romance dialects were not yet being put into writing.
