The JSON API is an advanced feature that enables developers to manage and access your site’s content via the REST API.
Enabling the Jetpack REST API allows developers to use the WordPress.com REST API for your site. You would likely only need this if another app or a custom application you build requires an API for your site.
To enable this feature simply go to the main Jetpack Settings page and enable the JSON API. After doing so, you should consult the app or developer with whom you are working for further instructions about how to use the REST API for your site.
This is not likely a feature you’ll need for regular use.