Is it legal to download books in PDF-format from https://www.pdfdrive.com/ ? How one can verify if the book given in the site is a legal copy or not?
You can't, in general, know whether a distributor of a work has permission to distribute, or is a pirate site. I verified that they have posted an illegal copy of a work that I created, and I know that I did not grant permission to them (or anyone) to infringe my copyright.
Both hosting and downloading works without permission is a violation of copyright law, so both parties are liable. Downloaders may erroneously rely on the "I didn't know!" defense, which in the US carries no legal weight. Even so, if you download my book, it will probably cost me vastly more to sue you for infringement than the damages that I might be awarded over your infringement. Usually, copyright holders go after the pirate sites, and only rarely go after particularly egregious serial downloaders.
3 of my books are up there without my nor my Publisher's permission. They've initiated legal action.
So no. It's not legal. Very little on the internet is "free", it is mostly stolen. Download only from reputable bookstores, Amazon at the very least.
This is a pirate site cloaking itself as a "public service".