What is the function of the Moderator band?
Does it mean it delivers impulses to the anterior papillary muscle faster?


Yes.


  If it does, how does this help? 


To facilitate the conduction time thereby allowing coordinated contraction of the anterior papillary muscle.


  and why only the anterior papillary muscle? 


This is because its chordae tendineae insert into anterior and posterior tricuspid valve leaflets, so only their contraction before systole would be optimum.


  What makes it special to have a moderator band?


For effective Cardiac Output because it helps in coordinated contraction of ventricular cells. The ventricles must maximize systolic pressure to force blood through the circulation, so all the ventricular cells must work together.