What was the backstory of the doubles in the film "Us"?

Here's what Red explains:

It was humans that built this place. I believe they figured out how to make a copy of the body but not the soul. The soul remains one, shared by two. They created the Tethered so they could use them to control the ones above. Like puppets. But they failed, and they abandoned the Tethered. For generations, the Tethered continued without direction. They all went mad down here.

source- https://www.springfieldspringfield.co.uk/movie_script.php?movie=us

So the shadows are the result of an abandoned government cloning experiment.

Red also says this:

When the girl ate, her food was given to her warm and tasty. But when the shadow was hungry, she had to eat rabbit raw and bloody.

The rabbits were food that was given to the shadows. Rabbits are easy to contain and they multiply quite quick providing for lasting food. The movie shows rabbits in cages adding to the fact that they were also part of the experiments.

Us is a good example of a film, where one cannot delve too deep into the logistics of their particular universe where said story is told.

Much like other films, such as A Quiet Place where one needs to ask the question eventually,

How quiet is quiet? Why is sometimes something loud, but other times something way louder isn't loud enough

Or in Star Wars, how come bombs fall downwards in space?

Or in the previous film by the same director of Us Get Out,

If the rich people who have the ability to transfer living brains into other bodies, how come no-ones ever heard of this, and why didn't they patent it and just become billionares...

It's not actually important to the story of how, and why and where the people in the bunkers came from. Just that they're there.

To answer your question, the people in the underground labs, were part of a country wide experiment to clone the 'surface' dwellers to test... ???

The underground labs are literally, giant huge government labs (again, nobody noticed building these?)

But how did they know who was going to meet who, and how did they time it all, and what about everything else?

Again not particularly important to the story, it's just 'stage' a set up a larger storey.

Bonus Answer

The rabbits are just a particular interesting symbology of the Director, also appearing in the movie Get Out.

To Quote:

“They’re an animal of duality,” he said. “They’re adorable but they terrify me at the same time. And they’ve got those scissor-like ears that creep me out.”

Peele, who also used rabbit imagery to chilling effect in Get Out, elaborated on the duality of rabbits in an interview with Rotten Tomatoes.

“Rabbits, they’re supposed to be cute. On paper, they’re adorable, lovable rabbits,” he said. “Have you ever gotten close to a rabbit? You can tell in their eyes that they have the brain of a sociopath. If you put a rabbit brain in a human body, you have Michael Myers, the killer. They have no empathy.”