PDF, the Portable Document Format, was introduced by Adobe in 1993 to solve one problem: a document should look exactly the same wherever it is opened, whatever the computer, the operating system or the fonts installed. It succeeded so well that PDF became the format for anything final: invoices, contracts, forms, manuals, tickets and papers. Adobe handed the format to the International Organization for Standardization, and it has been an open standard since 2008: ISO 32000-1 (PDF 1.7), followed by ISO 32000-2 (PDF 2.0) in 2017.
A page, not a text
A word processor file such as DOCX stores text as paragraphs that flow onto pages when the document is opened. A PDF stores the finished pages. Each page is a list of drawing instructions: this glyph at these coordinates, in this font; this line; this image in this box. Fonts are usually embedded in the file, which is why the layout never shifts.
That design has a price. A PDF does not necessarily know where a sentence or even a word begins and ends; it knows where each glyph sits. Text in two columns may be stored in an order that runs across both, and a hyphenated word at a line end is two separate pieces. Programs that extract the text have to reconstruct the reading order, and most of the time they get it right, but editing a PDF means moving objects on a fixed page rather than reflowing a paragraph.
Scanned PDFs
A PDF made by a scanner, or from a photo of a page, often contains no text at all: each page is a single picture of the paper. You can see the words, but no program can select, search or check them until the scan has been run through OCR (optical character recognition), which recognises the letters and adds a text layer to the file. If you cannot select a word in a PDF with your mouse, it is probably a scan without one. The Spell Checker runs that recognition itself, so a scanned PDF is checked like any other.
Variants for special purposes
ISO also standardises profiles of PDF for particular jobs: PDF/A for long-term archiving, PDF/X for professional printing and PDF/UA for documents accessible to people using screen readers. They are ordinary PDFs that follow extra rules, and any PDF reader opens them.
How the spell checker helps
You can open a PDF in the Spell Checker: its text is read out of the file and checked for spelling, grammar and punctuation, with style suggestions in Picky mode. When you have applied the fixes, the corrected text downloads as a Word document (DOCX) by default, because a PDF cannot be written back with its layout and you will most likely want to edit on; you can also choose a new PDF, ODT, TXT or another format.
Be aware of what that new PDF is: the corrected text, set as a fresh document, not the original page with its layout, fonts, columns and images. If the look of the original matters, use the check to find the mistakes, then correct them in the document the PDF was made from (usually a DOCX or ODT file) and export the PDF again. A scanned PDF without a text layer has nothing for a text checker to read until it has been through OCR. A single check covers a whole document of up to 150,000 characters.