Perform As User

If you don't specify anything here, the transition on the related will be performed under 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 which owns the necessary permissions. Usually this user account is assumed to be only technical (impersonation), with broad permissions, but not used to log into Jira by in life persons. 

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. For further example go here.