Why did the dwarves sing the song?

It was an old dwarvish song, presumably originating from a time not long after Smaug took over the Lonely Mountain. It describes a desire to return to reclaim their home under the mountain, primarily for the earthly treasure located there (the song describes in great detail the nature of the various treasures). Later verses in the song also briefly narrate the story of Smaug coming and destroying the town of Dale, forcing the dwarves to flee.

Here are the lyrics from the song as recorded in the book:

http://thorinoakenshield.net/misty-mountains-lyrics/

As to your question concerning why they sing it, I've always likened it to ancient Israelite songs of returning to Zion or American slavery spirituals -- such singing provides inspiration, motivation, and community, keeping a focus across generations on a cultural goal that holds deep, deep significance for a group of people. The song was likely ingrained in the dwarves' cultural heritage, perhaps being sung in their homes, their churches, their schools, etc. Because of this, when the dwarves are at Bilbo's house, ready to finally start this journey that all their people had been singing about for so long, they couldn't hold back, and that song seemed the best choice for communicating their excitement and sense of fulfillment.

They cut out a lot of the lyrics of the original for the movie. The abridged one in the movie is,

Far over the misty mountains cold.
To dungeons deep, and caverns old.
We must away, at break of day.
To find our long forgotten gold.
The pines were roaring on the height.
The winds were moaning in the night.
The fire was red, it flaming spread.
The trees like torches blazed with light.

Whereas the full lyrics are,

Far over the misty mountains cold
To dungeons deep, and caverns old
We must away ere break of day
To seek the pale enchanted gold.

The dwarves of yore made mighty spells,
While hammers fell like ringing bells
In places deep, where dark things sleep,
In hollow halls beneath the fells.

For ancient king and elvish lord,
There many a gleaming golden hoard
They shaped and wrought, and light they caught
To hide in gems on hilt of sword.

On silver necklaces they strung
The flowering stars, on crowns they hung
The dragon-fire, in twisted wire
They meshed the light of moon and sun.

Far over the misty mountains cold
To dungeons deep and caverns old
We must away, ere break of day,
To claim our long-forgotten gold.

Goblets they carved there for themselves
And harps of gold; where no man delves
There lay they long, and many a song
Was sung unheard by men or elves.

The pines were roaring on the height,
The winds were moaning in the night.
The fire was red, it flaming spread;
The trees like torches blazed with light.

The bells were ringing in the dale
And men looked up with faces pale;
Then dragon's ire more fierce than fire
Laid low their towers and houses frail.

The mountain smoked beneath the moon;
The dwarves, they heard the tramp of doom.
They fled their hall to dying fall
Beneath his feet, beneath the moon.

Far over the misty mountains grim
To dungeons deep and caverns dim
We must away, ere break of day,
To win our harps and gold from him!