The Yoast SEO Premium plugin (formerly known as WordPress SEO by Yoast Premium) adds several features to the Yoast SEO plugin. At the moment, the extra included functionality includes:

A redirect manager
This allows you to create and manage redirects from within your WordPress install, allowing you to easily redirect old URLs to new ones, and solve common issues. This is especially powerful because of the combination with the Google Webmaster Tools integration.
Built for scale, this redirect module can either handle the redirects for you or generate a redirect file that you can include in your Apache or NGINX server config.
Multiple focus keywordsWith this feature, you’ll be able to optimize a post for two completely different terms, or for two synonyms. This means you no longer have to choose between terms you’d like to rank for!

Example Video Tutorial

This is the first of the 13 video tutorials included in the Yoast SEO Premium plugin:

Installation Guide of Yoast SEO Premium

You can download the zip file mentioned in this post and upload it into your WordPress installation. To do this, go to Plugins → Add New and click the Upload tab:

When it's uploaded do not click "Activate Plugin" if you've already installed Yoast SEO. Instead, return to the plugins page to deactivate Yoast SEO first.

If you didn't have Yoast SEO installed or after you've deactivated Yoast SEO, click Activate on Yoast SEO Premium.

If you've updated the free Yoast SEO plugin, you can now safely delete this plugin after activating Yoast SEO Premium. Your settings will be safe.

Activate Your License

Go to SEO → Extensions → Licenses (tab) and enter your Yoast SEO Premium license key, this ensures plugin updates will work.

Go to the Licenses-tab which looks as follows. It lists all licenses for Yoast SEO Premium and all of its add-ons. Now enter your license key and press Save Changes (or Activate License, if available).

Your license should now be activated and the screen should look like this.

If you see the green notice, you've successfully installed Yoast SEO Premium!