Please make sure firstname.lastname@example.org is added to your safe senders list in your Amazon/Manage Your Content and Devices account:
And that your Kindle email address has been added to your NetGalley account (visit your NetGalley Profile, then Devices, and edit the "Send to Kindle" section to add your Kindle email address):
Once this is done go back to the book in your NetGalley account, click on the Send to Kindle button again:
If the book doesn’t appear on your device within ten minutes restart or sync your Kindle or Kindle app.
Click here for the full instructions for downloading books to your Kindle device or app.