I urge you to read these two articles about the "Noun Compounding" in Turkish if you are confused: http://ielanguages.com/turkish-noun-compounds.html http://www.turkishlanguage.co.uk/nouns.htm (search for "Turkish Compound Nouns" section)
Scroll up! "Hayvanat" means "animals " (borrowed from Arabic with Arabic plural ending) but it is used specifically for this expression (The Turkish plural of animals would normally be "hayvanlar".) and "bahçesi" means "their garden." (Literally Animals' garden, but think of Zoological garden or zoo.) Note how the possessive suffix -si goes on the possessed item.