What is the story behind Monument Valley?

The fragments of the story were written to help create an atmosphere for the game and are not narrative. In other words, there is no fully-worked out story that completely explains Ida, the crows, the monuments and the other aspects alluded to.

Ken Wong, Lead developer at UsTWo games is reported as saying

Wong called the game's themes more impressionistic and symbolic than narrative.

"The storytelling in the game is more akin to a song than a book or movie," Wong said. "The player can enjoy the game on whatever level they choose, and come away with their own meaning."

We're told straight out in the introduction to level 2: Ida is on a quest for forgiveness.

She's called a "thieving princess" in level 4 by the Ghost, implying she stole something. He tells us what in chapter 7:

Those who stole our sacred geometry have forgotten their true selves. Cursed to walk those monuments are they. Foolish princess, have you forgotten too?

And in Appendix 4:

When all sacred geometry is returned, so too shall be your crown.

And, most bluntly, the introduction to level 10:

In which at journey's end what was not meant for us is all returned

Ida's people, the crow people, have stolen mankind's sacred geometry, their "downfall" (level 9). Ida is on a quest to right her people's wrong and return the sacred geometry to whom it belongs. But, as the introduction to level 9 sadly proclaims:

In which there is nobody left to forgive us

Mankind was somehow wiped out. Ida is trying to do what she can to right that wrong, and pays tribute to their graves in this level. To one grave in particular - perhaps the Ghost's, the one character who frames the story?

The storyline might be a song, but at least the notes play a pretty melody rather than a random smashing of the keys.