Take down the item being held up by the anchor and remove the mounting screw. The toggle isn’t spring loaded, so removal of the mounting screw might permit the toggle to drop free. If that happens, turn the threaded anchor counterclockwise with the screwdriver and pull the anchor out.
Use your screwdriver to turn out the anchor as far as you can if the toggle is stuck to the wall. Use the needle-nose pliers to twist the head off the anchor’s body. Push the body back into the wall where it will drop away.
Use the screwdriver and a hammer to firmly tap the anchor into the wall if you can’t turn out the anchor at all. Keep tapping it firmly until you have tapped it through the wall so it drops away. Be careful not to tap so hard that you buckle the wall. Or you can just tap the anchor until it is slightly beneath the drywall surface and leave it in place to be patched over later.