Translation:You told me that you do not like them.
I think the answer key is misleading here.
Esperanto and English are different in how they treat indirect citations like this. In Esperanto (unlike English) puts direct and indirect citations in the same tense. The way you suggested is identical to:
Vi diris al mi, "mi ne ŝatis ilin."
You said to me "I did not like them"
You said to me that you had not liked them.
In English, when we say "You told me you didn't like them", the conversation usually goes like this:
Yesterday "I don't like them." Today: Why are you eating those - you told me you didn't like them.