cut down
刪 centers on removing or reducing something by cutting: cutting down, trimming, or deleting. The single given meaning 'cut down' captures this idea of elimination through a cutting action.
刪 combines 冊 (book, writing) with 刂 (knife), suggesting cutting or deleting text from a document. This is a standard semantic-component explanation.
A knife 刂 is used to cut down a stack of books 冊, trimming away unwanted parts. This visual directly shows cutting down or deleting.
For サン, imagine cutting down a sand dune with a knife: sand -> サン, and the knife slices through the dune to reduce it.