How would you explain this one to a DM?  It appears to be an enhanced skill with specific actions in combat. Does the advantage only apply if you are attempting to grapple, restrain, or shove?  Or does create an advantage for others if you are grappling/restraining or shoving the creature prone? 

My thought was you have advantage to start a grapple and once grappled advantage to pin or shove prone. If pinned (restrained) then others would get advantage on attacks too. I would probably use hitting cover even for a melee attack in this case so there would be some risk for the ally trying to stab at the foe in a bear hug. Same goes for others using melee attacks on the pined and/or prone foe.

The sticky point is how to handle the "pinners" follow on actions - the feat grants advantage to all grapple attempts but the restrained condition forces disadvantage for the pinner/attacker too so I would allow those with the feat normal attacks while restraining via a pin (a bear hug or knee to the back? - the combat narrative would be very important) to keep grappling but if they wanted to try and attack with a weapon they would have disadvantage. 

I find grappling and mounted combat to be severely lacking in rules and guidance.

Alright, that makes sense so advantage to grapple, etc. then advantage to pin (restrain), once you are restraining the opponent you lose this advantage while you are restraining because you can do little else but focus on your opponent.  Maintaining grapple is confirmed with a normal attack roll, attacking the opponent to further weaken in some way is done at disadvantage.  Other allies can attack with advantage but need to take grappler into consideration.  And the opponent now has disadvantage on dexterity checks because of the grappled position they are in.  I like that.  Unless I have it wrong, if so, please tell me.  

RAW might be a little vague to give DM more leverage for decisions, and thats what homebrew is for!

That's how I envisioned it - the dex check disadvantage was a key feature imo (I think restrained doesn't go far enough in this case) it forces the pinned foe to consider using strength to break the pin. It seems to make more sense when their arms are trapped or a big foot on their back.