"Я отвечаю на вопрос."
Translation:I am answering the question.
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It is a standard construction. The English verbs "look" and "listen" work in the same fashion: you have to say "look at" and "listen to" something, even though it does not make much sense in some other languages (for example, Russian agrees on "look" but keeps "listen" a transitive verb).
"I answer the question" isn't incorrect, but it is a bit unnatural. "I am answering the question" is how we would say it in a way that you want to be the only one or the first one to answer the question. Like someone else said in another comment thread, "I answer TO..." seems like you are answering to a person such as a boss, parent, or someone in charge of you. "I answer to no one" is the first phrase that comes to mind.