Google "Molly bolt" to see what you are dealing with. I have done all three in order of ease.
1) If they are recessed a little just patch over.
2) Drill the heads off with a drill slightly larger than the hole. and leave the rest in.
3) Put the screw in a few threads , then tap to straighten. Wiggle and pull it out.
I usually push them into the wall as well. A simple pop with a screw driver and hammer or you could screw in and pull out too. All will work, will still need to patch no matter what.
I have had nothing but bad luck with this type of anchor if you try to pull them out it usually takes a chunk of drywall with it. I have not tried drilling them out, but I will.
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