A PDF's pages aren't independent files stitched together — they're entries in one shared table that a reader walks through in order. Splitting means building a brand-new, minimal document around a single page's content for each page in the original, so each result opens on its own with nothing missing and nothing extra.
Because it's a structural copy rather than a re-render, a split page looks and behaves exactly like the corresponding page in the source file — same text layer, same links, same image quality.
Everything runs in your browser. Your file is never uploaded, never queued on a server, and never deleted from one — because it was never there.