Your field data, that is, the actual values you entered into the field for any product or other content, is stored in Shopify along with all of your other data. Specifically, the data is stored in Metafields, using a namespace particular to Custom Fields. This means that the values you enter into this app are available outside of the app, if you need them and have your own developers, using the Metafields API.
Your field definitions are stored in the app. Definitions are just your settings for the field, like if it's an HTML field and what you called it. The app uses these definitions to create the forms you use in the app, like the dropdowns or WYSIWYG fields.
If, for any reason, the Custom Fields app "goes down" your data will not be lost and your customers and visitors will not be impacted.