"The girl likes Philosophy."
Translation:Του κοριτσιού του αρέσει η Φιλοσοφία.
Remember that in Greek (similar, but not quite the same as in Italian), "I like" is actually articulated as "It is pleasing to me." Μου αρέσει is actually "it is pleasing TO me". So in this case "Psychology is pleasing TO her. (sorry for the caps - this forum doesn't seem to support italics or underlines). More commonly would by "Στο κορίτσι αρέσει η ψυχολογία" and duolingo does accept this answer. Understanding this construction also helps explain why "ψυχολογία" uses η and not την. It's not the object but the subject of the verb. It is pleasing ... to.. her.
The more complicated (for foreign language speakers) του κοριτσιού, του αρέσει says the same thing and του is repeated simply to imply: Psychology is pleasing to her, the girl. Her and girl are kept in the same case (genitive) because one is simply expanding on the other. Complicated, but I hope this helps.
αρέσω has a different transitiveness compared to English: it means "to like" but the one being liked is the subject! This is something that speakers of other languages might be familiar with (e.g. Italian mi piace, Spanish me gusta, Russian мне нравится): • (en) Juliet (subject) likes Romeo (direct object) • (gr) Στην Ιουλιέτα (indirect object) αρέσει ο Ρομέος (subject) or • (gr) Ο Ρομέος (subject) αρέσει στην Ιουλιέτα (indirect object).