Here's a very quick tutorial video about how you would deploy a Web API project on your local development machine to IIS:
Importantly he explains how to "Publish" the project, rather just building it. In your screenshot you're showing the build output of your project, which is fine to use as your "Physical path" when developing locally, but in production (i.e. on your server) you'll want to point IIS at the published path instead.
This shows how to do the setup on the server, view it over localhost & even how to reveal the app to your network by using your network private IP, but how you expose the API to the public is entirely up to you/your network admin.