to take something out and be rid of it: to pull a part out of the place where it sits
The verb is built out of two pieces: a common Greek verb for taking hold of something, and a prefix meaning "out of". The prefix does all the work here. The thing is not merely taken, it is taken out of where it was, and in the two verses that use the word this way it is an eye taken out of a head.
Both times the command comes in a pair: take it out, and throw it away from you. The second half shows how absolute the first one is. Nothing is being adjusted, bandaged, or covered over; the part that keeps causing the trouble is removed and disposed of. That is the whole force of the word, and it is why the saying is so hard to hear.
Greek had other ways to say something similar, and the difference is worth noticing. A verb for gouging or digging something out (ἐξορύσσω) puts the picture in the digging, and a verb for taking something off a person (ἀφαιρέω) puts it in the loss. This word puts it simply in the taking out.