How to remove shower drain piece (grip from inside)?
When I needed to replace the drain in my shower I was told to use this drain removal tool:



If that tool would work, there will be four cross bars in the bottom of the drain that the tool grips when inserted.  You take a screw driver or other rod and slip it through the holes on the side of tool to then turn it and remove the drain.

If the tool doesn't fit (which happened to me) an alternative is to superglue some scrap wood to the top of the drain as way to get leverage.  The down side is that method means replacing the drain rather than just putting it back.