> Presumably he used the same process, but substituting ZenDesk for Slack. Or else he used a different method that he doesn't describe.

He does describe the process.

> How does signing up for Zendesk using feedback@slack.com give him access to the target company's Zendesk instance?

Because like the author said "any one could sign up with any e-mail address and effectively read any support tickets created by that e-mail address."