If you don't specify anything here, the transition on the related issue will be performed as the same user who triggered this post function on the origin issue. Therefore that user must have the necessary permissions on the related issue.
In some restrictive setups, that user might not have the permissions on the related issue or even might not have access to the project of the related issue.
With 'Perform As User' you can specify a different user account that owns the necessary permissions. Usually, this user account is assumed to be only technical (impersonation), with broad permissions, but not used to log into a Jira account as a person.
In combination with the 'User is in Any Users' Condition, you can hide a transition from all other users than the 'Perform As User' user.