Obreht's novel is surprising in that it's about many different things, and that it builds metaphors for each of those things so adeptly. For a first time novel, that's kind of miraculous.
Obviously, the novel is about war. By never specifically naming anything except for a few town names when absolutely necessary, it's not about the Balkan war specifically but about genocidal war in general. And more to the point, "regular" peoples' behavior during and after such a war. She is trying, in an oblique fashion, to point out to us how culture affects and is affected by deep, running hatred of a different group of people.
As well, it is about the lives of those who lived to see the war happen, and all that transpired to make them who they are. The grandfather's tolerance, which makes him so different from the villagers around him; the apothecary's betrayal and the short-lived power that provides him; the characters who are forced to make a living from death of all kinds no matter the effect on themselves. We are who we are because of our circumstances, but very much also because of our choices.
The grandfather is the heart of the story because of that very thing. Obreht chooses to make each story (far past, current past, present) revolve around him. At times his character is too heavy-handed (such as his annoyance with Natalia for abandoning the zoo's tiger, or, when he forces Natalia to come watch the elephant).
What I can't quite figure out is exactly what the tiger is supposed to symbolize. Freedom? That is, from war, from society, from difficult choices. Power? His power holds an entire village in its sway, probably long after his death. Love? In his connection with the tiger's wife. Or even sacrifice? Since he ends up hurting himself perhaps to bring people closer together in a time of need. I hope it's all of these.