How do I remove glued-on carpet?
First, cut away the non-glued part with a craft knife or carpet knife.  Next, pull individual strands of the pile away from the mesh using a pliers.  Optionally, sand down the remaining bits of the carpet using your multi-tool.  Finally (and also optionally), patch any damage to the plywood subfloor with wood filler.

The last two steps are optional because it's in a small closet where nobody's going to notice any slight unevenness in the floor.