What are the steps that should be followed to develop and implement WebExtension APIs?
Please share some useful links.
A WebExtension API means some specific capabilities that a browser (the runtime environment for WebExtensions) exposes (through JavaScript) for use in extensions.
It's a bridge between the browser itself and extension code. It provides extra capabilities to extensions that are otherwise unavailable to regular JavaScript code.
To develop one? You have to change the browser itself. You can create your own fork of a browser, or go through the appropriate procedures to get the API specified, implemented, tested, and accepted as a contribution to the browser code.
I can't say for Firefox, but here's an overview of the process for Chromium: API Proposals.