In the movie Hobbit, The Desolation of Smaug, the band enter Mirkwood and are attacked by giant spiders.
In the movie the spiders appear to ‘chitter’ to one another. After Bilbo puts on the Ring, the chittering stops and he can hear them talking to themselves and each other.
My question is, is there any explanation in the Hobbit, LoTR, Silmarillion or other letters for the 'babelfish effect' when Bilbo puts on the Ring and understands the spiders?
I don't recall that happening the The Hobbit, but when Sam puts on the Ring in the Lord of the Rings, he is able to understand the speech of Orcs.
He heard them both clearly, and he understood what they said. Perhaps the Ring gave understanding of tongues, or simply understanding, especially of the servants of Sauron its maker, so that if he gave heed, he understood and translated the thought to himself.
This is from the "Choices of Master Samwise" chapter in The Two Towers. As far as I know, this is the best answer we get, and it seems like the narrator (or rather, the writer of the Red Book) is speculating, but it seems that the Ring does indeed have a "babelfish" effect on its bearer, as you put it. Thought it is presented speculatively, it does make a lot of sense.
When Sauron forged the Ring he put much of his "strength and will" and "inherent power" into the Ring (Of the Rings of Power and the Third Age). That's why destroying the Ring destroyed Sauron. And the understanding of the thought behind speech is one of the powers of the Ainur. In fact, it was something all rational beings in Middle-Earth were capable of. For example, when Elf Finrod first met Men:
Now the Eldar were beyond all other peoples skilled on tongues; and Felegund discovered also that he could read in the minds of Men such thoughts as they wished to reveal in speech, so that their words were easily interpreted.
Or when the Fellowship encountered Galadriel, she read their thoughts even without them intending to speak:
And with that word she held them with her eyes, and in silence looked at each of them in turn. None save Aragorn and Leogolas could long endure her glance. [...] 'Maybe it was only a test, and she thought to read our thoughts for her own good purpose.'
Even Denethor, a Man, was capable of doing so. Gandalf, speaking to Pippin, commenting on Denethor's shrewd questioning:
'He is not as other men of this time, Pippin, and whatever be his descent from father to son, by some chance the blood of Westernesse runs nearly true in him; as it does in his other son, Faramir, and yet did not in Boromir whom he loved best. He has long sight. He can perceive, if he bends his will thither, much of what is passing in the minds of men, even those that dwell far off. It is difficult to deceive him, and dangerous to try.
Tolkien discussed the ability behind this at some length in a manuscript published in the Vinyar Tengwar #39. But at any rate, since Sauron possesses this ability as well, and he poured his power into the Ring, it only makes sense that others wielding the Ring would also have their abilities in that area enhanced.
In the book, Bilbo only gets close enough to a group of spiders to hear their conversation after slipping on the ring. The invisibility conferred by the ring allows him to approach without being detected. But later he takes his ring off and can still understand the spiders:
He had taken off his ring when he rescued Fili and forgotten to put it on again, so now they all began to splutter and hiss:
“Now we see you, you nasty little creature! We will eat you and leave your bones and skin hanging on a tree. Ugh! he's got a sting has he? Well, we'll get him all the same, and then we'll hang him head downwards for a day or two.”
If the ring allows Bilbo to understand the spiders, this is not in any Tolkien canon, it's new in the Hobbit movies.
In the context of the movie it does make some sense. The trouble here is that the Hobbit is chock-full of talking (and otherwise intelligent) animals, whereas LotR is not (the spiders obviously talk normally in the book, and can also understand what Bilbo is saying). Trying to link the two and maintain consistency is a problem that must be solved: how do we go from a world where talking animals are commonplace to one where they are not? Tolkien chose to solve it by ignoring it (Bilbo is an unreliable narrator anyway which makes things easier). Jackson's solution is to have the Ring act as a translator; presumably via some form of mental communication.
Incidentally, there is one talking spider in the Silmarillion: Ungoliant, who assists Melkor/Morgoth in destroying the Trees of Valinor. We must assume that Ungoliant is some form of malignant spirit in spider form (Tolkien hints that she may have been a Maia: "some have said that in ages long before she descended from the darkness that lies about Arda, when Melkor first looked down in envy upon the Kingdom of Manwe, and that in the beginning she was one of those that he corrupted to his service" - presumably an incredibly powerful one) but there's no definitive final explanation. Either way this establishes some form of precedent for talking giant spiders that could have been used in the Hobbit movies, but Jackson is, of course, not legally allowed to use any Silmarillion material.
This is slightly late but I was recently rereading The Hobbit and it occurred to me that the ring might provide such powers for reasons besides just the spiders: when Bilbo was searching for a way to help the dwarves escape from the elven prisons in Mirkwood and the king's butler along with the chef guard just so conveniently happened to be speaking in plain English, allowing him to understand. It does strike me as quite strange: if they believed they were truly alone, wouldn't they be speaking in Elvish?
Giant spiders in book spoke in their peculiar manner:
,,Their voices were sort of thin creaking and hissing, but he could make out many of the words that they said."
What language they used it's a different matter. It might be that due to their nature as offspring of Shelob's brood (and thus demonic entity of great power Ungoliant, whose daughter Shelob had similar powers and abilities, even some sort of mind influence on Gollum, by the way Ungoliant spoke too), sentient creatures that are more demonic than simply huge bloated animals, they can speak in debased form of westron (better known as Common Speech, even trolls can speak, orcs are billingual, it's a common tongue in Middle Earth as the name implies).
While in The Hobbit Bilbo's magic ring wasn't yet conceived in Tolkien's mind as the Great Ring of Power, the One Ring of Sauron (though already there are certain common elements like the trickster nature, playing tricks on owner like putting itself on finger or coming off on it's own at most inconvenient moments). It is all nicely connected though once the Lotr was published so there is no disrepancy. So it might be retconned to the power of understanding the Ring gives the wearer (so the great spiders of Mirkwood might use form of Black Speech or any other form of language though that sort of speculation is unnecessary as I think they indeed used westron).
I also don't quite understand complains about various talking animals, while in fact there is very few of them that would speak with words in the manner of men (ravens of Erebor come to mind as primary example). Most use their own forms of language or communication and it is the members of sentient races men, elves, dwarves that have ability to understand or talk in the assorted animal languages (Bard has ability to understand and talk with trushes as ,,The Men of Dale used to have the trick of understanding their language, and used them for messengers to fly to the Men of the Lake and elsewhere", Beorn too talks with animals in their own languages, as well as he can somehow breed magical animals, much more intelligent, that serve him in his household, he even kept bees larger than thumb which is not natural in my opinion, dwarf Balin from Thorin's company could understand languages of birds though Bilbo heard only usual bird cries).
There is also category of unnatural creatures like Wargs (semi sentient but very bestial race of huge wolf-like creatures larger than ordinary wolves so much that orcs can ride them, when wargs would allow them) and Great Eagles (and great spiders count here too), that either talk in their own way like ,,dreadful language of Wargs" or were simply able to use language of other races.
I see no disrepancy either as in Lotr we have elves able to speak with trees, sense aura of places like some sort of emphatic ability, forms of telepathy or, to simply put it, magical creatures (like Mearas who understand speech of men though themselves do not speak as we do, but they are super horses in many ways), Dark Lord used various creatures of animal sort (birds especially) as spies, even in Silmarillion we have examples of an elf who could learn languages of animals (one of the sons of Feanor, Celegorm), Great Eagles of course too spoke in Lotr, one even sang to the people of Minas Tirith announcing the fall of Sauron (heard plainly by thousands of people in the city, including Faramir and Eowyn). Gandalf also spoke to the warg like things that attacked fellowship in Hollin so it implies they could understand him (though as it later turned out they were not typical wargs at all and it's not certain whether they actually were of this kind, as their corpses disappeared, possibly they were some sort of phantoms)
I think that if someone is looking for explanation then we can say that some creatures were simply enhanced and raised to higher level by the Valar (or Dark Lords) like maybe hound Huan, some are creatures that simply have their own ways of communication and various sentient people learn how to use them (call it magical ability if you will), and some might not be animals at all but things in such form (it is rather speculative but it might be that some of the ,,spirits from afar" sent ,,among the kelvar and the olvar" (plants and animals), these guardians like Yavanna envisioned the souls not of Ainur that formed new kinds of beings (like Ents, they breed and are alive, have souls but are not of the divine being Ainur).
It should be noted that in the movie one of the spiders (the one killed by Bilbo) spoke plainly after hobbit took off the Ring and thus he gave new name to his sword (the spider pierced by elven blade hisses ,,it stings" when Bilbo is already visible, without the ring, but in my opinion the scene of that event in book was much more awesome than this rushed sequence, Bilbo alone kills spider in brief and vicious struggle for life, then loses consciousness and declares after coming to and seeing spider's corpse ,,I shall name you Sting", even book battle with spiders was more glamorous at one point book-Bilbo throws himself from the tree branches to the ground to land in the cluster of spiders and starts slashing like a real warrior hero).
