If you don't specify anything here, the transition on the related will be performed with the same user who triggered this post function on the origin issue. Thus, that user must have the necessary permissions on the related issue.
However, in some restrictive setup, that user might not be allowed to do so on the related issue. He might not even see the project of the related issue!
With 'Perform As User', you can specify a different user account who has been granted the necessary permissions. Usually this user account is assumed to be only technical (impersonation), with broad permissions, and not used to log into Jira by any coworkers.
In combination with the Permission Condition in native Jira, or the User Is In Any Users Condition from JSU, you can hide a transition from all users that do not have the permission to execute it.