Printing a physical book is less forgiving than reading on a screen. A page that looks fine on your monitor can still hit a snag at the printer. The print-readiness check (sometimes called a preflight) is there to catch those snags before you export, not after.
What it does
The check looks over your book with a printer's eye and flags anything that might cause trouble, before you send files off to be printed. It is a gentle "have a look at this" rather than a wall you cannot pass. You get a chance to fix issues while it is still easy.
Why run it
Print-on-demand services have their own requirements, and a rejected file means waiting and re-uploading. Running the readiness check first means fewer surprises down the line: you catch the small stuff here, where fixing it is quick, instead of discovering it after an upload bounces back.
When to run it
Run the check when your book feels finished and you are getting ready to export for print. Give any flags it raises a look, tidy up what you can, and then export with more confidence. It is a small habit that saves real time at the printer.
Once you are happy, head to getting onto KDP and Lulu.