"His eyes are being opened."
I put this as my answer, would it be right also?
"i suoi occhi stanno essendo aperto"
Also can anyone explain what is going on in the suggested translation? What are the clitics referring to?
Your sentence should have been "I suoi occhi stanno venendo aperti"; venire is often used as auxiliary in passive sentences, and it must be for some conjugations. Putting it like this it implies that someone is opening his eyes; the given Italian instead implies that his eyes are opening (themselves).