(n) da, de, den,dan : they take the buffer letter -n- when added as a second suffix to an already extended noun.
So, in this word bahçesindeyiz " Si " is the first suffix and "de" is the second one, so we added a buffer "n" before it, thus, it will be "si-n-de" "...sinde...".
To learn more about suffix in turkish: http://www.turkishlanguage.co.uk/suffix.htm
In Turkish, nouns can take different endings depending on their role in the sentence. Subjects take nothing; the objects of most verbs take the accusative case ending -(y)I [y is a buffer letter added after nouns and the I represents vowel harmony]; if you want to say "in/at/on X" the case ending -DA is used, etc.
Basically it is just endings on nouns that change their grammatical role.
Because that's how Turkish functions.
The X Y (ex: the car key) and the Y of the X (ex: the key of the car) constructions require that you put 3rd person possessive on the second word.
- The car key = araba anahtarı
- The key of the car = arabanın anahtarı.
In a way, you're saying "Of the car its key".
Hayvanat bahçesi is the same. Literally: Animals their garden = garden of animals.
Sorry i forgot the last part of your question.
We at his zoo and we are at the zoo are the same in Turkish. Hayvanat bahçesindeyiz.
Your suggestion "bahçesisindeyiz" is logical, I admit, but it is incorrect.
If you want, you can add "onun".
Biz onun hayvanat bahçesindeyiz.