Drill a pilot hole for the hollow-core door anchor. Use a drill bit that is the same diameter as the anchors.
Press the hollow-core door anchor into the pilot hole until it is flush with the surface of the door. Use a hammer to lightly strike the head of the screw, which sits inside the hollow-core door anchor. This will set the teeth along the base of the anchor in the door surface material and provide friction against which you can tighten the screw.
Gently tighten the screw, using a screwdriver. By doing so, you are opening the metal wings of the hollow-core door anchor and causing them to squeeze the door material between themselves and the base of the anchor. Watch for the surface of the door to begin sinking inward. This will indicate that you have turned the screw too far and need to back it off a bit.
Remove the screw from the set hollow-core door anchor. Place the item you wish to hang into position over the anchor. Use the screwdriver to tighten the screw and hang the item.