Because you either say 'He eats neither chicken nor fish' OR you say 'He does not eat either chicken or fish' The either/or is used when you negate the main verb, but if you don't negate the main verb you have to put the negative in some other way, hence neither/nor. You can say 'He eats either chicken or fish' but that means exactly what it says. But 'He does not eat neither chicken nor fish' smacks of the double negative which works in some languages, but not in English.
Respect to whomever put that sentence together. I got it right, but I must admit it was a bit of a hit and hope in the end, lol. What's the betting I won't be so lucky next time :D
EDIT: Maybe an English for English speakers would be a good idea :-)
You are correct that double negatives are incorrect in English. "Neither... nor...", however, is an exception because both words are needed to complete the phrase. It is completely conventional and established English usage and in no way incorrect. You will find it in any grammar book.
Hi Maria, I think it's because there is a double negative in your sentence ("doesn't" + "neither...nor"). It should be: "he eats neither chicken nor fish" or "he doesn't eat chicken or fish"
The other option:
"he doesn't eat either chicken or fish"
sounds strange to me. I would tend to use "either... or..." in a positive sentence, although I'm not sure whether this is a rule:
"He eats either chicken or fish"