Enjoy your meal! in Russian is actually closer to the adopted from French Bon appetit! than to the English version, have a look:
– Прия́тного аппети́та!
Enjoy your meal! / Bon appetit!
Прия́тного is the the adjective прия́тный (pleasant, pleasing, agreeable, nice) in the Genitive case.
Аппети́та, as you might have guessed, comes from the word аппети́т (appetite), also in the Genitive case.
If you wonder why you need to use the Genitive case…
This phrase is actually a shortened version of:
– Я жела́ю вам/тебе́ прия́тного аппети́та!
[ya zhee-lá-yu vam/tee-byé pree-yát-na-va a-pee-tée-ta]
I wish you a nice appetite!
As in every Russian wish, after я жела́ю the Genitive case must follow.
But remember, in real life nobody says the whole phrase, only the last part of it: прия́тного аппети́та!
If you want to learn more about how to wish something in Russian, have a look at the following lessons:
Listen to "Enjoy your meal!" in Russian
The audio recording includes all the examples (in bold and blue) listed above.