Unscrew the nut and washer from the anchor using a wrench or drill attachment if necessary. Remove the head of the anchor from the threaded portion.
Use a hammer to tap the remaining section of the anchor into the surrounding material as deeply as possible -- tap gently to avoid causing excessive damage. Use a hacksaw to cut off any remaining protruding portion of the anchor.
Mend the remaining gap with an appropriate product for the material. Apply an appropriately colored brick-mortar compound to cover gaps in bricks, or, if the brick is badly damaged, simply cut it out and replace it with a fresh brick and mortar. Moisten damaged concrete and then apply a small amount of wet concrete or an expanding-concrete sealant.
Drill future anchor holes twice as deep as the length of the anchor. This will make it easier to embed the anchors when they are no longer needed, typically removing the need for hacksaw cutting.