Several people have asked why "answers to" isn't correct.
In English, to "answer to" is to acknowledge the authority of someone: "the bishop answers to the pope". In contrast, "answers" simply means to respond to something.
Someone's father may well be in a position of authority over that person, but (and here I CONJECTURE) I believe this sentence is using "answers" in the general, not authoritative, sense. The `a is common after a verb in French indicating the direction of the action, so it is fair to mentally translate it to "to", but in this case literally adding a "to" in English changes the meaning of the verb.
Sorry, you could say "He responds to his father." or "He replies to his father.", but the most common English way to say it is "He answers his father." The verb "answers" just is not followed by "to" for this meaning. If you put "to" you will change the meaning, see above posts by shriramk and northernguy.
Be careful with "répondre" as it is often best translated as "to answer". "To respond" means something different as it means to react. You could say something back and not answer the question and you would still be responding. Sometimes "répondre à" does mean "to respond to" as in "He has always responded to my kindness." Il a toujours répondu à ma gentillesse. http://www.larousse.fr/dictionnaires/francais/r%c3%a9pondre/68435?q=repondre#67687 http://dictionnaire.reverso.net/francais-anglais/répondre
If you look up "respond" in Larousse it gives variations of "repondre" to cover "to respond to" and "respond to" in its examples. On the other hand translating "repondre" to English, there is barely a mention of "respond". So is "He is responding to his father" correct? Duo says no, but then says "He is responding his father" is correct which is a phrase I don't think I have ever heard.
French Verb Tenses and Moods (Simple and Comound) + all other forms of a verb + examples. http://learn-french-free-with-stories.weebly.com/brief-explanation-of-the-french-verb-tenses-and-moods-simple-and-compound.html