Install and activate
After purchasing Gato AI Translations for Polylang, you will have access to the corresponding plugin file. Download it and install it in your WordPress site (under Plugins > Add new > Upload), and activate it.
Upon activation, an admin notice will request to provide the license key:

Click on the link to go to the Settings page. Then copy/paste the license key, and click on Activate licenses (or Deactivate/Validate).

The plugin will now be activated and enabled:

Installing the plugin on a staging site
The staging site (hosted under localhost, .dev, .test, etc. domains) requires the same license key as the live site, however it does not consume a license activation.
Updating the plugin (when a new version is released)
While your purchased product's license is active, you will have access to all plugin updates.
You can download the latest plugin via My account, or directly update it in the WordPress Plugins page.