In English that sentence means "I am replying to everyone's questions".

In English you reply TO a question, which is exactly what на means here. Всем is dative case, which is exactly what "them" is in English, but as the dative case is otherwise practically extinct in English, you have to say "I reply to them". In Russian "I reply them" is sufficient. "I reply to them to questions" would sound very awkward, English prefers a genitive "their" or "everyone's". Which Russian also can do, but doesn't have to.

The English verb "answer" works differently, but "reply" illustrates better what is going on here.