"My father has the keys."
Translation:Ο πατέρας μου έχει τα κλειδιά.
That's one reason why the second τα is there :)
It's a short pronoun, so it can only be an object (direct or indirect - here direct since it's an accusative pronoun rather than a genitive one). Subjects don't have short-form pronouns.
This lets you distinguish between e.g.
- Το αγόρι το βλέπει το κορίτσι. - The girl sees the boy.
- Το αγόρι βλέπει το κορίτσι. - The boy sees the girl.
The first is literally something like "The boy, him sees the girl".
Even in a sentence such as this one, where the singular verb form έχει precludes having plural τα κλειδιά as a subject, it's clearer if you add a resumptive short-form object pronoun before the verb.
This form is rather colloquial indeed. In a formal text you probably wouldn't encounter it, you'd read "Ο πατέρας μου έχει τα κλειδιά.". This form could be found in literature or in speech, and it's used to put more emphasis on the object (in this case the keys) rather than the subject (my father). For example, you could hear a conversation like "-Μπορείς να μετακινήσεις το αυτοκίνητο; -Όχι, τα κλειδιά τα έχει ο πατέρας μου" (-Can you move the car? -No, my father has the keys). We don't really care about the person who has the keys - the father- , the important think is that the keys are unavailable.