what's your process for searching engines work?

Is this for explaining to a client, or are you trying to fix something specific? For the explaining kind, the library comparison lands with most people: crawlers are the ones collecting the books, the index is the catalogue, and ranking is the librarian deciding which ones to hand you first. Most "why isn't my page showing" problems turn out to be the book never making it into the catalogue.

Three stages, and it helps to keep them separate in your head. Crawling: bots follow links and sitemaps to find pages. Indexing: the page gets fetched, parsed and stored - this is where a lot of pages quietly fail, because of a noindex tag, a robots.txt block, or the engine deciding it is a duplicate of something else. Ranking: when someone searches, matching pages are pulled from the index and ordered on relevance and quality signals. When a page is not showing up, work backwards through those three. The URL Inspection tool in Search Console tells you which stage it is stuck at.