Why does the poet break lines into more than one in the 3rd and 4th stanzas of "Postcard from Kashmir" by Agha Shahid Ali?

Kashmir shrinks into my mailbox,
my home a neat four by six inches.

I always loved neatness. Now I hold
the half-inch Himalayas in my hand.

This is home. And this the closest
I'll ever be to home. When I return,
the colors won't be so brilliant,
the Jhelum's waters so clean,
so ultramarine. My love
so overexposed.

And my memory will be a little
out of focus, it in
a giant negative, black
and white, still undeveloped.

I assume when you say "the poet breaks the line into more than one" you mean the poet breaks the sentence into lines.  An important part of poetry, the line is completely different than a grammatic sentence.  A full sentence can continue for more than one line, and sometimes more than one stanza.  The reason for breaking a sentence into lines of poetry can be varied, some of which you identify in your question.  Poets can break lines for rhythmic or sound effect, as in a poem with a specific rhyme scheme.  A line can also be broken to achieve a certain shape, as in concrete poetry, to emphasize a word or phrase, to compare or contrast words, or to force the reader to speed up or slow down.

Agha Shahid Ali's beautiful little poem, "Postcard from Kashmir" mimics its subject matter in size and shape.  The poem, only four stanzas, is like a postcard in that it packs much into a small space.  In this way, you could argue that #1 is the correct answer, and that the poet is trying to recreate the look and feel of a postcard.  But as in most poetry, there is no right or wrong answer, only interpretation.

I actually see a much stronger argument for #2.  Shahid Ali has broken the line at some very significant places in stanzas 3 and 4 which affect the reading of the stanza.  In particular in stanza 3, there is a high amount of repetition of phrase structure: "so brilliant...so clean/so ultramarine...so overexposed."  By breaking the lines where he does, Shahid Ali draws attention to this repetition, which creates a subtle rhythm that is almost prayer-like or chant-like in quality.  He also draws attention to the language of photography by ending stanzas 3 and 4 with the words "overexposed" and "underdeveloped."  By drawing emphasis to these words, Shahid Ali compares the permanent quality of a photograph to the impermanence of memory.

Why does the poet break the lines in the 3rd and 4th stanzas of "Postcard from Kashmir"?

Kashmir shrinks into my mailbox,
my home a neat four by six inches.

I always loved neatness. Now I hold
the half-inch Himalayas in my hand.

This is home. And this the closest
I'll ever be to home. When I return,
the colors won't be so brilliant,
the Jhelum's waters so clean,
so ultramarine. My love
so overexposed.

And my memory will be a little
out of focus, it in
a giant negative, black
and white, still undeveloped.

There could be a variety of reasons why Ali chose to use enjambment in the third and fourth stanzas of "Postcard from Kashmir." Out of your three choices, the least likely is three. Enjambment is a literary device used in poetry that affects the flow of the piece and the words/phrases that are focused on. Hardly ever will a poet use it to make reading "convenient."

The second option is the most likely: force the reader to notice specific phrases. Enjambment is a good technique to use when you do not want to use bold or italics for emphasis. Because of the natural pause that arises when you finish a line of a poem before moving on to the next, the reader is left focusing on the last phrase or word of that line. For example:

This is home. And this the closest
I'll ever be to home. When I return,

The sentence continues to the third line, but let us focus on these two lines. The first line emphasizes the word "closest" because of where Ali broke the sentence and created a new line. The reader is focused to feel the emotion left hanging by the first line and embrace the importance of that feeling in the context of the poem.

Your first option, a visual representation, could be correct for some poems. However, in Ali's case, there is not really a shape to be made. That said, even though it is not the best choice, enjambment is a technique that is also used to make a poem more visually appealing on the page. Although it is not being used in Ali's case to create a shape, it is used to prevent long lines from extending from one side of the page to the other.

The breaking of the lines into more than one is called in the technical terms of literary criticism either "enjambement" (a French word used in English) or "run-on line". An enjambement splits the sytanctic unit so that the last syllable of the line does not correspond with the grammatical break. The line and thus its meaning overflow into the next one. The effect on the reader is to communicate the emotional flowing of the speaker's thoughts and to convey his nostalgia for a place where he will not return in the near future. Although the speaker would like to return to his Kashmir home, he can only see it reproduced on a postcard. He is just as torn apart from it as words are torn apart from each other in the poem. The syncopated rhythm of the last two stanzas makes the thoughts of the poet appear "out of focus" and in that quick succession typical of highly emotional state of minds.