beantworten
The verb »beantworten« is a transitive verb. This means it needs an accusative object. It would be wrong to use a transitive without an accusative object:
wrong: Ich beantworte.
correct: Ich beantworte die Frage.
I answer the question. I'm answering the question. I'll answer the question.
This accusative object tells you what you answer. In most cases you'll answer a question, but you can also answer a letter:
Ich beantworte den Brief.
I answer the letter. I'm answering the letter. I'll answer the letter.
antworten
This verb is not transitive. Even stronger: It is an intransitive verb. (There are verbs which are neither transitive nor intransitive. They feel well without an accusative object, but you can also use one with them.) You can use »antworten« without any object if you want and it still is a grammatically correct full sentence:
correct: Ich antworte.
I answer. I'm answering. I'll answer.
Here is a situation where you would use it without any object:
The host of a quiz show has asked a question to a candidate, and the candidate has the choice to either answer the question, or to skip it. So, the host asks:
Werden Sie diese Frage beantworten oder überspringen Sie sie?
Will you answer this question or will you skip it?
If the candidate wants to answer it, he has these possibilities:
Ich beantworte sie.
I'll answer it.
Ich antworte.
I'll answer.
Of course this is also possible:
Ich antworte auf die Frage.
I'll answer to the question.
But the point is, that German grammar does not force you to use this prepositional object. A prepositional object is never mandatory for grammatical reasons. Maybe you need it for semantic reasons, but if you have a sentence with a prepositional object, then the sentence without it is also always grammatically correct.
Also note, that »antworten« can not take an accusative object. This it what defines it as intransitive verb:
wrong: Ich antworte die Frage.
wrong: Ich antworte den Brief.
(Just for completeness: An example for verb that is neither transitive nor intransitive is »trinken«. You can use it like a transitive verb »Ich trinke das Wasser.« but also like an intransitive verb: »Ich trinke.«)
free dativ
The part »dem Mann« in »Ich antworte dem Mann« is an optional dative object. There are many forms of dative. Some of them are bound to the verb which means, that they are mandatory:
correct: Der Hut gehört dem Mann.
The hat belongs to the man.
wrong: Der Hut gehört.
The hat belongs.
But there are many situations where you can add a dative object to a sentence that also would be complete without this additional dative object. If this is the case, we call it a free dative object. And this is the case in the sentence
Ich antworte dem Mann.
I answer to the man.
Compare it with
Ich antworte auf die Frage.
I answer to the question.
In English you use the same construction for both sentences, but the meaning is different. The part »to the ...« encodes two very different things. In one sentence it's the addressee of the answer, in the other it's the cause of the answer. In German you use a free dative for the addressee but a prepositional object for the cause. And you can combine them in one sentence:
Ich antworte dem Mann auf die Frage.
I answer to the men to the question.
But In English you would prefer this construction in this case:
I answer the question to the man.
But this is in German
Ich beantworte dem Mann die Frage.
This sentence is interesting, because it demonstrates, that a free dative also can be used in a sentence with a transitive verb. »Dem Mann« is an optional free dative, »die Frage« is the mandatory accusative object of the transitive verb »beantworten«.
Maybe even better English is
I answer the man's question.
But now you say, that the question belongs to the man (he is the owner of the question) which is in German
Ich beantworte die Frage des Mannes. Ich antworte auf die Frage des Mannes.
So now you have a genitive attribute inside the accusative or prepositional object.