If you are using Publish for the first time you need to install the app on the airtable base you want to us it on.
You can find the app in the airtable marketplace or here.
Once the application is installed you are prompted to input your API key. If you are using the free plan you do not need to enter an API key and can click the button to continue.
If you have signed up to a paid plan you can log in on the baseboosters homepage and click the "Get API Key" button in the top navigation.
Once you have done this bit you are ready to use Publish!