Why are "answer me" and "answer the question" acceptable but not "answer me the question"? Is it similar to "explain me (something)"?
Sometimes it actually is ok to “answer me a question”. Acceptability may vary. For example, this sounds fine to me:
Here are several examples from Google Books, all of which sound find to my ear:
The simple answer is because we no longer talk that way. Here’s a rather famous quote that uses the very thing you have asked about:
As you might detect, that is a valid but archaic formulation. (Another sign of deliberate archaism there is the present subjunctive following ere.)
The OED has several transitive senses of answer, but almost all are marked either archaic or indeed even obsolete, especially the bitransitive ones.
Here are a few such examples from the OED:
The more complicated answer is a lot of people don't accept that answer can be ditransitive (i.e. - they don't think it can validly modify both me and the question in OP's example).
I'm not going to say they're wrong - I'm perfectly happy with I'll open you the door, for example, but fairly obviously most people aren't.
But I would say that in the case of [in?] question, it's worth noting there are apparently over half-a-million written instances in Google Books of...
...so if it is wrong, a lot of people either don't know that, or don't care. I think many verbs are "marginal" in this respect, and that answer is considered "ditransitive" by a significant minority.
For reasons that aren't fully clear to me, people are more prepared to "stretch" marginal verbs when the additional indirect object is me. So *"I answered Bill the question" sounds bad to people who don't mind "Answer me that!"
I think it's probably connected with the fact that forms like "Give me the money!" are acceptable to all because the indirect object (me) is a recipient. When you say "Answer me that!", what you really mean is "Give me an answer to that!". Semantically, answer serves as a "giving" verb, allowing it to be used ditransitively.
The specific indirect object me probably figures disproportionately high in the total number of utterances a person makes (we're always saying variants of "Give me this!" and "Give me that!"). My guess is that predisposes at least some of us to accept "Answer me that!" more easily than *"Answer Bill the question!"
The simple answer is that, while answer is a transitive predicate, and therefore has a direct object (referring either to the question that was answered or the addressee of the answer -- but not both at the same time); nevertheless answer is not a bitransitive predicate, and therefore has no indirect object.