Poet : Nobel Laureate Mr. William Butler Yeats
As I begin with explanation of my Understanding (which has evolved out of reading the poem, thinking over it, reading about the abstracts from various other writers and reading about various concepts and things mentioned in the poem), first I would like to talk about a bit of background:
Background:
Mr. William Butler dedicated this poetry to Harry Clifton (Henry Talbot de Vere Clifton, a poet from his era) who had gifted W.B. yeats an eighteenth century Chinese carving in Lapis Lazuli (Lapis Lazuli is a blue coloured semi-precious stone which was mainly found in Afghanistan and Persia) on Yeats 70th birthday. It was a scene representing a mountain, paths, trees, a temple (or a small house, that's still not clear) at the half way and three men, who were about to climb the mountain. Picture is shown below (Courtesy Wikipedia):

Looking at the stone and the picture carved on it, Yeats Imagines, that when these three old men, (out of which, first two walking ahead should be the masters and the third one walking behind them should be their servant, who is carrying the luggage and a musical instrument), will reach that small house/temple at half way up, will sit there and look down at the world, then they will observe the rise and fall of civilizations, upcoming wars and the triumph of art, music and literature over these tragic events through their old, matured and happy eyes.
Understanding:
Intro:Yeats first talks about the bad things happening around and the role of art, music and literature in still keeping spirits high and focusing on the good things as rising and falling is part of life, failing doesn't mean we stop living, and then he discusses lapis lazuli painting and his imagination of three men watching this whole drama in the last two stanzas.
Lapis Lazuli
--------------
Stanza 1:
I have heard that hysterical women say
They are sick of the palette and fiddle-bow.
Of poets that are always gay,
For everybody knows or else should know
That if nothing drastic is done
Aeroplane and Zeppelin will come out.
Pitch like King Billy bomb-balls in
Until the town lie beaten flat.
Explanation 1: As this poem was written by Yeats in 1938, just before World War II (1939-1945) had begun, thus here he says that, those hysterical (mentally not sound, in this context, from the experiences of first World War) women are even worse than the destruction's in our present world, as they don't support painters (represented through word "palette"), musicians (represented through word "fiddle-bow") and poets, who always talked about optimistic things through their art and tried to spread peace. These women believe in politics than arts and believe that everyone should fear a certain war, when aeroplanes and zepplins (a flying object used for transferring people and bombs, used in WW1 by Germans) [aeroplanes and zepplins are representing war] will bomb the whole city (as had happened in UK during WW1) and the whole city will be left in ruins (until the town lie beaten flat)
Stanza 2:
All perform their tragic play,
There struts Hamlet, there is Lear,
That's Ophelia, that Cordelia;
Yet they, should the last scene be there,
The great stage curtain about to drop,
If worthy their prominent part in the play,
Do not break up their lines to weep.
They know that Hamlet and Lear are gay;
Gaiety transfiguring all that dread.
All men have aimed at, found and lost; [18]
Black out; Heaven blazing into the head:[19]
Tragedy wrought to its uttermost.[20]
Though Hamlet rambles and Lear rages,[21]
And all the drop-scenes drop at once[22]
Upon a hundred thousand stages,[23]
It cannot grow by an inch or an ounce.[24]
Explanation 2: Unlike those hysterical women, the artist(Characters) like Hamlet, Lear, Ophelia, Cordelia...although are playing tragic roles, they do not stop acting and start weeping on the on going tragedy. They stand tall, perform their duty (which is to act) and complete the play until the curtains fall. Here yeats is attaching real life with stage, where he is comparing those messy hysterical women who were depressed and afraid with these characters, who are conveying a message to continue with good, irrespective of how difficult the time is. "Gaiety tranfiguring all that dread", i.e. the goodness and determination of these characters is converting that bad to good, their goodness is a representation of their high spirit and passion. {for lines 18 till 24}Although the characters who are portraying/playing the tragic roles like Hamlet or Lear where in there roles they might see extreme tragedy, but still that tragedy (which is represented as "It", in the statement, 'It cannot grow by an inch or an ounce.') cannot grow on them even a bit. They get over these material emotions as they even perform it.
Stanza 3:
On their own feet they came, or On shipboard,
Camel-back; horse-back, ass-back, mule-back,
Old civilisations put to the sword.
Then they and their wisdom went to rack:
No handiwork of Callimachus,
Who handled marble as if it were bronze,
Made draperies that seemed to rise
When sea-wind swept the corner, stands;
His long lamp-chimney shaped like the stem
Of a slender palm, stood but a day;
All things fall and are built again,[35]
And those that build them again are gay.[36]
Explanation 3: This stanza talks of the rise and fall of Old civilizations which were first Invaded by different dynasties and people, who came from different other places on various modes of transports, then they were destroyed and then later on they were again built by those men who believed in Optimism. Yeats gives examples, that even the art of Callimachus is also now gone, like the civilizations and their wisdom, which later end up on rack, just like the sea waves, which rise and fall. {Line 35-36} things (represents nearly everything, be it dynasties, civilizations, countries....etc) fall , but they are again built and here they are built again by people who believe in living and are optimistic.
Stanza 4:
Two Chinamen, behind them a third,
Are carved in Lapis Lazuli,
Over them flies a long-legged bird,
A symbol of longevity;
The third, doubtless a serving-man,
Carries a musical instrument.
Explanation 4: Now we come to the last two stanzas for which the poem is titled. The carving on lapis lazuli features a scene wherein three Chinese men are climbing up a mountain. A long-legged bird flies overhead, and Yeats tells us that bird is “A symbol of longevity.” Also, one of the Chinese men, Yeats assumes, is a servant, because he carries a musical instrument and is walking behind other two men.
Stanza 5:
Every discoloration of the stone,
Every accidental crack or dent,
Seems a water-course or an avalanche,
Or lofty slope where it still snows
Though doubtless plum or cherry-branch
Sweetens the little half-way house
Those Chinamen climb towards, and I
Delight to imagine them seated there;
There, on the mountain and the sky,
On all the tragic scene they stare.
One asks for mournful melodies;
Accomplished fingers begin to play.
Their eyes mid many wrinkles, their eyes,
Their ancient, glittering eyes, are gay.
Explanation 5: Now is the detailing of the Picture drawn on that Lapis Lazuli, where yeats imagines, the discoloration as the Snow on Mountains and small cracks and dents as the path traced by huge amount of sudden flow of water. Then at half way, he sees that house, with a tree, which yeats is not sure of, whether its a plum or a cherry tree, where those three old china men are climbing towards. Now as mentioned, yeats imagines them to be sitting at that small house (yeats delights , means he imagines them to be feeling comfortable there and at peace) and stare at the mountains and sky and see all those tragic scenes through their wrinkled eyes (i.e. they see all this but they are in Transcendence ) as they listen to the mournful music played by the servant.