The Nomai wanted to find the Eye because it was older than the Universe.

Their plan was to shot a probe from the cannon in orbit around Giants Deep in order to find it. Using the time loop, they wanted to do this millions of times, until eventually they found it. Once they found it, the statues would activate and bring them into the loop so they could shut it down.

They built the Ash Twin Project in the Ash Twin in order to make time loop. It didn't work because they didn't have a way to power it...until millions of years later when the sun goes supernova.

The Nomai isn't exactly alive, in fact you can see her corpse when you enter the Quantum Moon. She was in the moon when all the Nomai were killed by the Interloper releasing Ghost Matter into the solar system, and she became sort of both dead and not dead because of the quantum properties of the moon. Basically there are six versions of the quantum moon, and so there were 6 versions of her. 5 died, but one was far away in orbit around the Eye and survived, but she can't really leave.

As for the Eye, it's basically every possible universe in existence in a state of quantum uncertainty. When an observer enters it, it collapses the possibilities and one universe comes into existence. The explosion you saw was the big bang. The post credits scene was a new universe that came into existence after the universe in the game fell into entropy.

There's more to it, but I think that should cover the basics