What should I do with the ZIP file?

The free widget and the pro are two separate widgets. The pro widget is not an update from the free widget.

Pro widget installation:
Upload the ZIP file with WordPress (Dashboard > Plugins > Installed Plugins > “Add New” button > “Upload Plugin” button >) or copy it to your plugin folder ([your-wp-installation]/wp-content/plugins/term-and-category-based-posts-widget) to your WordPress installation.