She likes me translates to: "Je lui plais."    Why isn't the pronoun  "elle" ?

Hi Mary,

In 'Je lui plais', for 'She likes me' , lui could replace both a man or a woman. You would know which from the context.

It is an indirect object pronoun which has to be used because the verb is 'plaire à '.

Have a look at the following lesson for more details-

Using lui/leur = him or her/them (French Indirect Object Pronouns)

Hope this helps !