The suffix umuz is added to kus to form kusumuz (our bird). Then, as this noun kusumuz (our bird) is being described by the postposition "yaninda", the posessive particle, in this case "un" is added to the noun kusumuz to form "kusumuzun", which out of context means our bird's.
Extra: In postpositional phrases it turkish, it seems to me that the noun being modified by the post position is almost seeming to posess the postposition, despite the postposition not taking the genitive case, which is required for showing possession.
Slight modification to your comment: Type 1 postpositions (including "yan") use a possessive suffix, while the nouns that possess the side/top/etc take genitive case.
kuşumuz (our bird) + un (genitive case ending)
yan (side) + ı (possessive suffix) + n (buffer) + da (locative)