You must have a Premium Individual or Premium Family account to get your free Google Home Mini. If you aren't a Spotify Premium member already, you can sign up here (it's ~$10/mo for individuals).
Click the Get your Google Home Mini link.
This page will automatically generate a discount code for you and redirect you to the Google Home Checkout page.
Order your Mini and wait for it to arrive! Shipping is free, too. :)
If you have friends or family who use Spotify, share this guide with them so they can claim their free Google Home too!