In the BP_Stoplight_Variable_RepNotify Blueprint's Event Graph (pictured below), you will see that the script calls for only the Network Authority to handle the light switching logic. The Streetlight Scalar variable representing the state of the light is specifically marked as RepNotify, so both the Client and Server have the OnRep_StreetlightScalar function called when the value changes.
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The OnRep_StreetlightScalar function then handles updating the visuals of the streetlight such that it works on all machines.
Setting a variable to RepNotify can be done through the Details panel of the variable under the Replication drop down menu.
With this setup, both Client and Server see the same sequence executed for the streetlight when playing in game.