Hi!
The answer is d. : Elle y joue avec lui parce qu'elle aime gagner.
The pronoun "y" is used to replace a group of words following the verb starting with à/au/aux, just like in this case: "aux dames" (meaning "checkers").
The three other answers are all wrong, as the pronoun "en" can't be used to replace "aux dames". It is used to replace group of words following the verb starting with de/du/d'/des/la/les (example: J'ai besoin de ce livre --> J'en ai besoin)
(Also, the answers b and c do not make sense : b. means "she plays because she hates checkers" while c. means "he plays because he is really tall").