How can one start as a book reviewer?
From the publisher's point of view the point of these reviews is to sell more copies of the book. Therefore you need to be one of these:


A well-known authority in the field. It's especially good if you're affiliated with a well-known university like MIT.
Some kind of well-known personality. For example if you have 100k Twitter followers, that would work. Or maybe you host TV shows like "The Planet", that would work too.


If you are one of these then feel free to write to the publisher offering to review books. If your review is complete early enough it's possible they will add your comments to the book cover; otherwise it'll mostly be in the promotional materials. One thing to note is that if your review is poor, it might simply end up unused.

Alternatively, you could write to the editors of magazines that publish book reviews, such as Physics Today or Scientific American. It's possible they just need the text and are less concerned about who is writing it (you will still need some credentials though). Being neutral 3rd parties, they can publish negative reviews.