Specifically the last lines of the Wasteland:
Datta. Dayadhvam. Damyata.
Shantih shantih shantih
The poem was written in 1922, and the invocation can be taken as a response to the horrors of the first World War, but I'm interested in the mythological context, and wonder what insight may be found in the specific text Eliot was referencing.
Looking at Swami Krishnananda's book on The Brihadaranyaka Upanishad (emphasis mine):
This instruction, which was communicated to the Devas, Manushyās and Asuras – gods, men and demons – by the single letter Da repeated three times, meaning Dāmyata, Datta, Dayadhvam – be self-controlled, be charitable and be compassionate, is applicable to all mankind.
So, this is from the the Brihadaranyaka Upanishad, chapter 5, second Brahmana (one of the more recent ones, looking at the dates given in Wikipedia). If you're new to this, know that many tales of Hindu mythology are about the struggles between Asuras and Devas. Swami Krishnananda says Asuras "are extremely cruel in their nature," "[…] always insult, injure and harm other people," "[…] are hard-hearted people." They're the demons, but not all Asuras are necessarily evil. The antagonist of the Ramayana, Ravan, or the Asura king Mahabali, are considered capable rulers, pious, wise and knowledgeable. However, the constant warring between Asuras and Devas, while good for creating stories, isn't good for the normal people caught underfoot.
Accordingly, in this Brahmana, Lord Brahma, the Creator in the Hindu Trinity, instructs Devas to show restraint (in enjoying pleasures), Asuras to be compassionate, and mankind to be charitable. Mankind has the qualities of both Asuras and Devas, and so mankind should follow all three instructions.
Now, if these three has actually followed these instructions, so many wars could have been averted. If mankind would embrace self-control and compassion, we wouldn't be warring all the time.
Shanti literally means peace, and the triple utterance of shanti is common (see this post on the Hinduism Stack Exchange), so that by itself is not from a particular Upanishad.
To add to muru's excellent answer, taking on the "why" part of the question, this final part of Eliot's poem presents us with a world in ruins, not as much in substance as in spirit. The prevailing intention I think, is to depict spiritual emptiness: with the invocation of the biblical crucifixion with no resurrection, the depiction of the Grail with towers upside down in air, everything hints to a world failed in its deepest spiritual meaning. The pilgrim of the lonely path is projecting his ghost, making three with the narrator that by all appearance is also a part of the same being, creating this silent, thirsty, decomposing trinity. And then, the Thunder speaks, to each and all, as in the Upanishad: "What have you given?", "Why are you locked inside yourself?", "Take control". As the thunder's voice is heard (where in many religious writings, including all semitic, to have heard the holy word is to understand and accept and live by it), the world recomposes itself, and the trinity, now as one, is seen fishing peacefully.
So in all, I see it as a metaphor of what the world is without giving, compassion and control, and how he who hears the veddic teachings will find his peace in this decomposing world. Why is the text referenced? Simply because the whole section is nothing but the poetic illustration of this very text from the Upanishads.
As muru mentioned in his answer, the Upanishad Eliot is referencing is the Brihadaranyaka Upanishad. Eliot added the following footnote:
“Datta, dayadhvam, damyata” (Give, sympathise, control). The fable of the meaning of the Thunder is found in the Brihadaranyaka—Upanishad, 5, 1. A translation is found in Deussen's Sechsig Upanishads des Veda, p. 489.
I am quoting this note because there are a few issues with it, as Harish Trivedi pointed out in Colonial Transactions: English Literature and India (Manchester University Press, 1995, page 126):
As we all know, this [“Datta, Dayadhvam, Damyata”] comes from the Brihadaranyaka Upanishad, even if some commentators on Eliot and/or their copy-editors still cannot spell Brihadaranyaka correctly and consistently). Many critics have pointed to the puzzling fact that the order in which the three injunctions occur in the Sanskrit, “Damyata, Datta, Dayadhvam”, has been reshuffled by Eliot to become “Datta, Dayadhvam, Damyata”, but as we do not have even a persuasive surmise much less a reasonable explanation as to why he should have done so, there is perhaps some reason to suspect a little private trick on Eliot's part, a little harmless pedantic joke.
Harish Trivedi also points out that Eliot's footnote sends readers to the wrong part of the Upanishad, namely Chapter 5 section 1 instead of Chapter 5 section 2. For a translation of the relevant section, see for example The Upanishads by Max Müller:
The threefold descendants of Pragâpati, gods, men, and Asuras (evil spirits), dwelt as Brahmakârins (students) with their father Pragâpati. Having finished their studentship the gods said: ' Tell us (something), Sir.' He told them the syllable Da. Then he said: 'Did you understand?' They said: 'We did understand. You told us "Damyata," Be subdued.' 'Yes,' he said, 'you have understood.'
Then the men said to him: 'Tell us something, Sir.' He told them the same syllable Da. Then he said: 'Did you understand?' They said: 'We did understand. You told us, "Datta," Give.' 'Yes,' he said, 'you have understood.'
Then the Asuras said to him: ' Tell us something, Sir.' He told them the same syllable Da. Then he said: 'Did you understand ?' They said: 'We did understand. You told us, "Dayadham," Be merciful.' 'Yes,' he said, 'you have understood.'
(For commentary on this section, see for example The Brihadaranyaka Upanishad by Swami Krishnananda. Swami Krishnananda points out that the three types of beings understand the same syllable differently depending on their nature.)
Harish Trivedi also points out that Eliot's footnote sends his readers to a German translation, which would have been out of reach for many of his readers, instead of an English one.
Finally, Harish Trivedi points out that Eliot may have been attracted to a rhetorical device:
This is the device through which this single syllable ["Da"] evokes and expands into three independent and disparate words and thus, what was obscure becomes explicit, what was cryptic becomes lucid, and what was elementally incoherent becomes humanly discernible.
In conclusion, Eliot's footnote provides some eludication because it explains or translates “Datta, dayadhvam, damyata”. However, the reader is sent to the wrong section of the Upanishad, possibly intentionally, since some of Eliot's other footnotes are less than helpful (e.g. "The following lines were stimulated by the account of one of the Antarctic expeditions (I forget which, but I think one of Shackleton's): ...."), as if Eliot were actually parodying the concept of footnotes itself.
