In a poem, an author has fewer words and a different structure than in prose.  Poets rely on figurative language to get their meaning across and create the sound they want.  Here are some examples from your poem.

Alliteration (the repetition of consonant sounds)- Alliteration does not create meaning, but it does make a nice sound.  It can assist in the rhythm of a poem.  There is an example in line 2-3, where the b and d are repeated.

Croaked away in Bingle Bog.

Every night from dusk to dawn (lines 2-3)

Rhyme and couplets (pairs of rhyming lines)- Rhyme is also used to help a poem sound a certain way.  It can also help with meaning, because it can create a sing-song, almost silly rhythm.  This poem is arranged in couplets.

Neither stones nor prayers nor sticks,

Insults or complaints or bricks

Stilled the frog's determination

To display his heart's elation. (lines 11-14)

Allusionis a reference to history or literature.  In this poem, we have an allusion to “I’m Nobody, Who are You?” by Emily Dickinson.  The poem comments that it would be dreary to be a frog and “to tell one's name – the livelong June –/To an admiring Bog!”

Dumbstruck sat the gaping frog,

And the whole admiring bog (lines 19-20)

Satire is when an author or poet makes fun of a subject or comments on it humorously.   In this poem, the frog tricks the nightingale into thinking he can teach her and therefore destroys her.  The frog is repulsive, and the nightingale is beautiful, but the frog is the one who wins.  Although this seems like a sweet children's poem, there is a strong message here about manipulation of the strong over the talented but weak.

Please explain the literary devices (figures of speech) used in the poem "The Frog and the Nightingale" by Vikram Seth.

One literary device in "The Frog and the Nightingale" by Vikram Seth, is the author's use of rhyme, shown in the poem's rhyming couplets. The poem is made up of pairs of lines that rhyme with each other—a rhyming couplet. An example is shown below. Note that the last word of the first line rhymes with the last word of the second line ("frog" and "Bog").

Once upon a time a frog

Croaked away in Bingle Bog

Seth uses onomatopoeia as well, which describes the sound it stands for, like the "hiss" of a snake or the "buzz" of a bee. It is seen with the word "croak" ("She was startled by a croak") and "clapped"...

And the whole admiring bog

Stared towards the sumac, rapt...

And, when she had ended, clapped…

Alliteration (the repetition of the same consonant sound found at the beginning of words in a group) is included. Note the bolded letter "t" at the beginning of each word. This device is based on the identical sound:

Toads and teals and tiddlers...

Consonance is based on the same concept as alliteration, but the repeated sound is a consonant found within or at the end of a word—that is part of a group of words. In the following example, we can find consonance with the "t's" or the "r's," but I have highlighted the "s's" at the end of three words:

...stones nor prayers nor sticks

Assonance is almost the same as consonance, but refers to the use of vowels—with the same sound. For example:

...you who

The poem is an example of personification, giving animals human capabilities: to talk, criticize, and even write. The frog states:

And, of course, I wield my pen

For Bog Trumpet now and then.

There is repetition with "awn and awn and awn," and with "Did you… did you…" Generally repetition is used to stress an important piece of information, but it also may be used if a poet requires a certain number of beats in a line, which relates to the poem's meter.

Vikram Seth also cleverly provides a parody on names with the following lines, which are similar to the "Earl of Sandwich" and the "Duke of Kent" (both historical figures), and the "Count of Monte Cristo," a fictional character.

Owl of Sandwich, Duck of Kent,

Mallard and Milady Trent,

Martin Cardinal Mephisto,

And the Coot of Monte Cristo...

We can see that the poem is an allegory, which is defined as...

A symbolic narrative in which the surface details imply a secondary meaning.

The poem relates the story of a frog who sings poorly in his bog. When competition arrives in the form of the nightingale, the frog pretends that he his more important than he is, takes advantage of the bird's inexperience, exploits her, criticizes her, and eventually drives all the joy for singing out of her—because he takes advantage and she does not have enough self-confidence to resist. Ultimately, she tries so hard to be what he wants her to be that she dies. The deeper meaning parallels people who take advantage of others who are less experienced in life, and how some people become targets for these unscrupulous sorts because they lack self-confidence. For the above reasons, the poem is also a satire, exposing "social predators" of this nature.

Finally there is also irony. After the bird dies, the frog criticizes her, saying she should have known:

That your song must be your own

The irony is found in the frog giving such good advice when all along he pushed, prodded and forced the bird to sing his song—her music, his way, so her song could not be her own any more.