The phrase “to be skinned out of one’s eyeteeth” means: To be right royally hornswoggled, bamboozled or flimflammed, that is, cheated or deceived, and there are twenty or more less polite ways to express the unfair methods employed in such skulduggery.
Whereas the expression “to cut one’s eyeteeth” indicates that one has reached (presumably) years of discretion, “to have one’s eyeteeth drawn”, or the American version, “to be skinned out of one’s eyeteeth”, conveys the reverse impression, to be duped, to be made a fool.
It is the hayseed taken over by city slickers or, conversely, the man or his wife coming from a big city to a country village who is most likely to be skinned out of his eyeteeth.