Remove the screw from the anchor
Grip the anchor with needle-nose pliers
Twist and pull the anchor straight out
If it will not budge, gently pry around the edge with a flathead screwdriver
Use a utility knife to cut paint or drywall compound around the anchor if needed
Push the anchor into the wall cavity if it is a hollow wall anchor and cannot be removed cleanly
For plastic expansion anchors, pull the collar outward while twisting
For metal anchors, collapse the anchor inward with a screwdriver before pulling
Fill the hole with spackle or wall patch compound
Sand smooth after the patch dries
Repaint if needed