Simplified path. The puzzle showed the main steps; the full route is being checked.
Old Italian
riso
‘The grain’
into Anglo-French
Anglo-French
ris
‘The grain’
into Middle English
Middle English
rys
‘The grain’
became English
English
rice
‘The grain’
The story of rice
Old Italian riso passed into Anglo-French as ris. Middle English used rys before the modern spelling rice.
These are selected later stages, not the word's ultimate origin. Merriam-Webster traces it further through Greek to Iranian origins; that earlier route is outside this question.