Translation:He says that he is a model but he is really a priest.
Yes. Egentligen means actually, really in the sense of contradiction, like in the example above where what he says contradicts what he is. Verkligen, on the other hand, is used to mean really, truly.
Du har verkligen lyckats med målningen = You have really succeeded with the painting.
Does that help, sort of?
This sentence is actually very useful in a situation where you try to warn a Swedish-speaking friend that the person he/she has encountered in an establishment serving refreshments might not be of the profession he claims to be. And, the weirder the sentence, the more essential it is to know it in the situation where things can only go even more weird. (I have not yet encountered a Duolingo sentence for which I can't imagine some usage situation, maybe it tells something about how my mind works...)
I put: He says he is a model but he is really a priest. (missing the 'that' after 'says'.) It marked it as incorrect, even though it is grammatically correct and has the exact same meaning. on a related note, would you also be able to drop the 'that' in Swedish for a less formal sentence, like we can in English?