"Who likes eating?"
Translation:Qui aime manger ?
"Aimer bien" translates to "like". If you were to say "Je t'aime" to someone, it has a romantic quality to it, and translates to "I love you". If you were to tell someone that you like them as a friend, as in "I like you, you're a good dude", you would have to say "je t'aime bien", as to not weird your friend out ;) So when it comes to foods/actions/colors/etc, both "aimer" and "aimer bien" mean you thoroughly enjoy that thing, and both are acceptable and correct. But when you are talking about a person, unless you want them to think you are in love with them, use "aimer bien".
Why would you insert "a"? It doesn't really has to do anything with the original sentence; maybe you tried to translate "likes to eat" to "aime 'a' manger"? In which case the word "to" is unnecessary in French, in English it's used before a verb to indicate the infinitive time, in French the infinitive time verbs have their own single word. I hope it helped.
There are some french verbs that pretty much always take the preposition "à", such as "désobeir à", "demander (quelque chose) à ", "faire attention à", "parler à", and many more.. All of these, of course, need an indirect object. In English, this "à" can translate to a number of different prepositions. In the case of "who likes to eat", it is like MiguelB5 said-- "to" in "to eat" simply indicates an infinitive, it is NOT a helping preposition. It would be the same if you were to ask the question: "Where to eat?" (Where are we going to eat tonight?). Simply: "Où manger?" would suffice.