Getting a discussion tab on a github repository?

If you're looking for per-repository discussions like in the link you mentioned, that's a new feature that is in a limited beta at GitHub. It will be rolling out to more repositories and projects soon and GitHub will provide more information when it's available more widely.

Update Dec 2020

You can enable it from project settings.

Old Answer:

As of Oct 2020, private repositories are not eligible for the Discussions beta.
Repos must:

1. Be public visibility
2. Have at least 3 contributors

Source: https://github.community/t/can-one-apply-for-github-discussions-beta/121120/332

As of the 9th of December Discussion tabs are now rolling out to all GitHub’s repos just go to settings to were you can turn off and on; issues and wikis etc, at end you will see the new discussion tab option view a screen shot

https://github.community/t/can-one-apply-for-github-discussions-beta/121120/550

It seems that at the moment team discussions can only be created if there is a team/an organization, that seems to imply that private/personal repositories do not have a discussion. For creating a discussion with a team/in an organization you can currently refer to https://help.github.com/en/github/building-a-strong-community/creating-a-team-discussion.

As of today 7th May 2020, GitHub, talked about team discussions in their recap of their satellite conference from yesterday, link to recap of team discussions

Dec - 2020

For Private free account is does not seem possible to get the discussion tab on the actual repo

On the free plan to add a discussion you need to first create a team, then attach a team to a repo. Your discussions will not show in the actual repo. In order to access the discussion you need to click on the main account and then click on teams, then click on the team.

For the reason above I advise you to ditch GitHub for an interface that makes sense.