could you perhaps explain the significance of fili and kili's death in the hobbit book?
I love this question because the answer is very much not as distinguished as most people would hope it to be.
The thing about the deaths of Fíli and Kíli in the The Hobbit is that they don’t really matter much at all. The knot of The Hobbit isn’t the Quest of the Dwarves to reclaim Erebor, but Bilbo’s change as a Hobbit (or, as a person and character in general). On the second page of the first chapter we see that “This is a story about how a Baggins had an adventure, and found himself doing and saying things altogether unexpected” (10). Gain and loss is a major theme in the work, which is explored through characters doing things that are either presumed or never presumed of them. As the narrator goes on, “he gained—well, you will see whether he gained anything in the end” (ibid).
As any astute reader of The Hobbit will notice, Fíli and Kíli are sparse characters all throughout the text. We get their names and the color of their cloaks, but that's basically it. We see that they nearly drown near the start of the adventure due to mishaps with the ponies, and maybe have a word in edge-wise now again. Fíli and Kíli don't matter much: they are periphery characters that add to the depth of the story without needing to go into detail.
When Fíli and Kíli die the text becomes very medieval, and very jumbled. There is a shift in tone and lexicon, most likely because the narrator is now taking from other eye-witness accounts, as Bilbo is unconscious. Based on how the Company is grouped together by narrative influence, we can guess that this section of the text is either taken from the POVs of Gandalf of Balin. The characters of the Company with the most to say, and who are most commonly interacting with the scenes, are Thorin, Bilbo, Gandalf, Balin, and sometimes Fíli and Kíli.
So, when we read "Fíli and Kíli had fallen defending him with shield and body, for he was their mother's elder brother" we see the shift (303-4). This is information that would not have been known to Bilbo, but is known to either Balin or Gandalf. It's made more certain that this is reported by either those two because they are both present in the final chapter of the text, discussing with Bilbo the tale of the Quest. Perhaps this is when Bilbo is penning his book, and perhaps they are filling in some gaps in his memory (since he did have a tendency to be unconscious at the most inopportune times).
Because the story is about Bilbo and Bilbo's change as a character, and because Bilbo most likely did not know of Fíli and Kíli's relation to Thorin (or really any of the Dwarves' relations to one another), it becomes even more clear that their death is simply a matter of warfare. It is tragic and upsetting, because they were the liveliest and (perhaps) youngest of the Company, but it it wasn't significant that they died. What's significant is that Bilbo is changed, for he must abide by the rules of Perilous Realm. Once you traverse into Faerie, you do not come back the same, if you come back at all.
There are some medieval touches that can be seen in the way the death scene of Fíli and Kíli was constructed. To me, this passive sentence reads very much like a line out of an Icelandic family saga, or the Eddas of Saemundr the Wise or Snorri Sturlusson. It is brief and factual, like the reckoning of a family line that is the kernel of family sagas, and truly the beginnings of Old Icelandic literature. The trope of the maternal uncle is also very clear in this scene, when we learn that Thorin is their "mother's elder brother." Maternal uncles are a prominent plot figure in family sagas and Indo-European myth cycles. They are linchpins of texts, either vetting or misleading characters with advice, or simply by their blood relations.
There is also the the trope of "cleansing the sickness" from a family line. Thorin is not the only descendent of Dúrin, as we can see by Dáin's ascension to the throne; that wouldn't be possible unless Dáin also had a "divine" right to rule, based on his ancestry. But Dáin also doesn't have the gold-sickness, the dragon disease that Thorin was privy too. And if Thorin fell to the sickness, wouldn't Fíli and Kíli also fall to it? We do know that the younger Dwarves were against Thorin's brash actions at the siege gate, against the Elven King, but we don't see much of how they act otherwise. Like everyone else in the company they pick up golden harps and dance among the gold once Smaug has left: no one is the wiser. Killing the heirs along with the king isn't an uncommon medieval trope. It is tragic and it is harsh because his line is ended, but, again, it's not the heart of the story.
It's somewhat difficult to admit that their deaths didn't matter, but textually this is true. To Bilbo's story, it is unimportant to his plot that periphery characters died alongside the king--for why wouldn't they? It would be unbelievable if only the tainted king died in the battle. So their deaths become a part of the "fictive dream" that we require.
It also rings with a deeper narrative, a story outside of Bilbo's own: the story of the Dvergatal, the "Catalog of the Dwarves" in the Poetic Edda. Many scholars (including myself) debate about what the purpose of this list of Dwarf names is, seeing as it is "superimposed" (argued by some, but not by me) in the midst of a poem for no "apparent" reason. The names must mean something if they are placed in a spot near the end of the creation of the universe but also near stanzas that talk about the destruction of that said universe. Those Dwarves did something to make their names go into the mythological codex. Tolkien might have spied them and thought that perhaps some died heroically, perhaps some found the lost kingdom of the Himinbjorg. By Norse tradition, which the Dwarves are based and enmeshed in, Fíli and Kíli die in a manner preferable to warriors. You are slain in battle, valiantly, instead of cowering, instead of losing only to live and be debased. In modern terms, we'd say something like "going out with a bang." And for the Norse, this a great thing. It's the desired means of departure.
So, while the deaths of Fíli and Kíli aren't so much significant, there are dozens of interweaving strings that make the thick cloth of their textual depth. Tolkien wasn't lazy when he wrote their deaths, as they are complex things that connect to myths and cultural texts all over the northern region of Europe, even reaching back into Indo-European tropes. But, he was careful enough not to overshadow the place that he set for himself and Bilbo when he started the text. The Hobbit is a story about a Baggins having an adventure, and being changed from it. What happens along the way is something that's part of a much greater culture and scheme, one that comes to life the more the reader becomes acquainted with the medieval texts that Tolkien held so dear.
Tolkien, J. R. R. The Hobbit, Or, There and Back Again. Boston: Houghton Mifflin, 1966. Print.
