Hello, the plugin is capable of running in localhost–that’s how I do all of my development. However, it must be able to reach the Font Awesome API server at https://api.fontawesome.com (there is no way for you to run that locally). If it can’t successfully reach the API server, it will show that error message that you saw.
As for loading resources from the Font Awesome server, there are two different ways that things are being loaded from Font Awesome servers:
(1) The plugin loads metadata from api.fontawesome.com. This is only for configuring the plugin, though, so it would only be loaded when an admin user is using the settings page. And even then, there is some caching involved to optimize these loads. The plugin does not have to make requests to the Font Awesome API server on normal page loads. It’s the plugin’s back end (PHP) code running on your WordPress server that makes a request to the Font Awesome API server.
(2) The plugin adds <script> or <link> tags to your page templates, depending on how you’ve configured it. Those tags will load resources from one of the Font Awesome CDNs, depending on whether you’ve configured the plugin to load Free or Pro, and whether you’re using Kits. It’s the browser (not the plugin) loading a given page that loads the resources from the CDN. And because of caching optimizations here as well, those requests will often just re-use the browser’s local cache instead of making new network requests for assets.
As for the slow experience you’ve mentioned, I would need more information about what behavior is happening more slowly than might be expected. On the server? In the browser? On front-end page loads? Or admin settings page loads?