Great news! A LISTEN NOW button means that any NetGalley member, when signed into their account, will be instantly approved for that audiobook. Just press the Listen Now button and you'll see the audiobook added to your Shelf, and your NetGalley Shelf app, too.
Some publishers opt to enable the LISTEN NOW feature for audiobooks that they want to promote widely, or for special content like catalogs and samplers. You can see a full list of LISTEN NOW audiobooks in the catalog by going to Find Titles, and then under Browse click Listen Now. You can also click on any Category page to find Listen Now books in a specific category.