Translation:When Gamze apologized to you, I also wanted to apologize to you.
I looked up 'dilemek' and its used in auxiliary verbs, do all of them require this case or is this one special ? like başarı dilemek,baş sagligli dilemek
özür = forgiveness
dilemek = to wish for
You can think of özür dilemek as "to wish/to ask for forgiveness". That way you know that you have to ask for forgiveness from someone and not apologize to someone.
Remember that not all languages work like English. Some verbs may directly happen TO an object, but in other languages some verbs happen FROM an object. And also, the meaning of the words are technically different from their English counterparts which accounts for the difference.