How to remove plastic anchor from brick?
Pulling it out with long nose pliers is one option, but it often comes out in pieces if you do that. 

If you want to pull it out with a screw, you need to engage the screw enough to grip the plastic, but not enough to make it expand. This position may not exist, or if may be further out than you think. Sometimes a screw engaged just 2-3 turns will give you something to get a grip on with pliers or a claw hammer (not under the screw head but near the tip). A very slightly undersized screw can help with some fixings. 

Drilling it out completely is tricky. Once it breaks into pieces they tend to go round in the hole rather than drill. 

It looks to me, from the fact that there's still some sticking out but it won't pull out, that the hole is too tight (drill too small). The universal plastic fittings often call for different hole sizes for different materials.