"Please answer the meaning of this word."
Agreed. This literally means "the intangible concept of the meaning behind this word has asked you a question, please reply to it" which is meaningless. Honestly I prefer to learn languages by literal translations ("this word's meaning, give answer please") but Duolingo usually doesn't, so I would expect "please give the meaning of this word" or something.
Because one can't provide a response or an answer to a meaning, we must either change 'answer' to some other word, or, as SSSRoaB has opted, change the structure of the sentence somewhat, so that we can keep 'answer.' I like your suggestion to add 'with' after 'answer,' but language teachers and literally minded translators often frown upon structural change even more than word substitution.
Adding 'with' shifts the focus from a direct object to the modality, how one is to answer or do something: With [or using] (a hammer), by [means of] (this algorithm), in (English). In Japanese, such things are usually expressed by X で (i.e. で instead of を, which is reserved for the direct object), X を使[つか]って (/X を利用[りよう]して), X を以[もっ]て.