Hello ecki02, it is a widget, and every widget in wordpress works in the same way. There area different ways to add a widget, but a good start is go to Appearance ? Widgets in yout admin panel.
Manual installation
Download the zip file and extract the content
Upload the folder weather-in-any-city-widget to your WP plugin folder /wp-content/plugins/ directory
Go to Plugins > Plugins, and activate the plugin
Widget
Go to Appearance > Widgets, and drag the widget to a sidebar
Configure your widget
Shortcode
Go to Appearance > Widgets, and drag the widget to a sidebar
Configure your widget
Copy the shortcode generated at bottom
Remove widget
Paste shortcode wherever you want
Please, this plugin is a free service. Reconsider your evaluation. Thanks.