Why is "piacciono" applicable here? I would usually use "Mi piace" for I like. I assume it has something to do with the subject of the liking being a plural but my usually guess would be that "piacciono" means "they like". Any suggestions to figure this kind of thing out in future?
In the English sentence the subject is "I". In the Italian sentence the subject is "le donne"(plural). This is more evident if you change the sequence of the words "le donne mi piacciono".
English also used to have dative expressions like this - Chaucer is full of them. Two which survive (although perhaps not in everyday language) are "woe is me" (woe is to me) and "methinks" (it seems to me)