Learn Turkish: Bu böreğin içinde peynir var. - There is cheese inside this börek.

There are two things happening here. First, to say something is "inside" an object in Turkish, you literally say it is in the "inside of" it. This requires the genitive case, so we add the -in ending to mean "of the börek". Second, because we added a vowel to a word ending in k, a rule called consonant mutation changes the k to a ğ to make it easier to pronounce.

How exactly does the word içinde break down?

It is built from four parts: iç (inside/interior) + -i (possessive suffix matching böreğin, meaning "its") + -n- (a buffer letter) + -de (locative suffix meaning "in/at"). The buffer -n- is a special rule: it is always required when you add a spatial case ending (like -de, -den, or -e) right after a third-person possessive suffix.

Could I just say "Bu börekte peynir var" instead?

Yes, you absolutely can! Bu börekte peynir var simply means "There is cheese in this börek." Using böreğin içinde is just a bit more specific, emphasizing that the cheese is stuffed right on the inside.