The first thing I would do to troubleshoot this problem is to create a new comp, add a shape layer with the Rectangle tool, select the puppet pin tool, turn on view mesh and try and add and move some pins. If that works then there is something going on in the layer on the comp you are having problems with. A layer with puppet pin applied cannot have any transform animations or an animated alpha channel.
If a new comp with a only a shape layer and puppet pin does not work, and you are running the latest build of your version of AE, reboot, and try again. It that fails, try changing the Project Settings > GPU setup. If that fails, reset your preferences.