I have cloned the repository XM-Cloud intro, and deploy it successfully in my local docker. I haven't configured anything yet for Edge so i am assuming Edge is not being utilized yet. Now the question is, in this case, where Edge is not available, where exactly publishing is happening and how sites are still available for public user as neither we have web and nor edge?
Sitecore Experience Edge is a cloud-based CDN, it's not something you run locally. In basic terms, it is a CDN which is used as a replacement for your CD instances and Web database. You can read more about it on the docs site: https://doc.sitecore.com/xp/en/developers/101/developer-tools/sitecore-experience-edge-for-xm.html
When running XM Cloud locally using Docker as you mentioned, Edge is not involved. You're locally running the Sitecore CM instance and the Rendering Hosts. The Rendering Hosts run against the CM instance and Master database, so there is no publishing action to be performed when running locally in this way.