Translation:The girl likes to eat cheesecake.
Indeed, distinguishing between the personal pronoun "του" and the possessive "του" can sometimes be tricky in written speech, but in this case you need to pay attention to the fact that "αρέσει" needs a personal pronoun as an object in genitive.
There are cases, however, where it really is impossible to distinguish between the two. Then, and in such cases only, we normally add a stress mark to the personal pronoun precisely to make the meaning clear (even though it is a one-syllable word). For example:
-Το κορίτσι του είπε την αλήθεια: His girl told the truth.
-Το κορίτσι τού είπε την αλήθεια: The girl told him the truth.
-Το κορίτσι του του είπε την αλήθεια: His girl told him the truth (no stress mark here as it's clear which pronoun is which)
Of course, in real-life situations, you can usually rely on the context as well!
When you use ''αρέσει'' with a noun, it sounds quite unnatural to simply use the noun in genitive. We therefore use the personal pronoun as the verb's object instead, with the noun acting as a 'definition' ( I'm not sure about the term here) of the pronoun (so it must be in the same case as the pronoun). Alternatively, you can use the noun with a preposition: ''Στο κορίτσι αρέσει να...'', which is actually a better expression especially when it comes to formal/written speech (Του κοριτσιού του αρέσει is common, but rather informal).
Note that the same is true of indirect objects in genitive: they are usually personal pronouns, or they are replaced by nouns with prepositions (e.g. Του έδωσα το τηλέφωνό μου/Έδωσα στον Γιώργο το τηλέφωνό μου)
Repeating a pronoun is very common in Greek and sometimes the fine line between what sounds natural or unnatural. There is another wording you can use, "Στο κορίτσι αρέσει...", where you don't have to repeat the pronoun because you have used a preposition (Σε+το).
Come to think of it, it's the same in Spanish, which doesn't have grammatical cases, however. "A la chica le gusta comer cheesecake".
I see below the comment that this verb αρεσει has to have a personal pronoun, as well as the noun in the right case. I am having trouble finding out what other verbs do this.
Also, could you say (and apologies for no accent): του κοριτσιου του του αρεσει να τρωει τσιζκεικ his girlfriend likes to make cheesecake? Or would Greek prefer να τρωει τσιζκεικ αρεσει στο κοριτσι του?