Learn Polish: Proszę pana, dlaczego pan czyta cudzy dziennik bez pozwolenia? - Sir, why are you reading someone else's diary without permission?

If you used swój, the sentence would mean he is reading his own diary! The adjective cudzy is the opposite of swój—it specifically means "someone else's" or "belonging to others".

What case is cudzy dziennik here?

It is in the accusative case because it is the direct object of the verb czyta (reads). Since dziennik (diary) is a masculine inanimate noun, its accusative form looks exactly the same as its nominative (dictionary) form, and the adjective cudzy matches it.

Why does the sentence start with Proszę pana if the speaker isn't asking him for a favor?

While proszę usually translates to "please" or "I ask", the fixed phrase Proszę pana is the standard, polite way to say "Excuse me, sir" or to respectfully get a man's attention before talking to him.

Why is pozwolenie (permission) in the form pozwolenia at the end of the sentence?

The preposition bez (without) always forces the noun that follows it into the genitive case. Because pozwolenie is a neuter noun, its ending changes from -e to -a in the genitive.