We don’t know what happened right at the big bang, but we can start from a few fractions of a second afterwards.
The big bang created particles, all crushed together in a tiny space. That space started expanding, giving room for the particles to join together as atoms, and then those atoms to form molecules.
Those molecules then clumped together to form clouds, and bits of those clouds clumped together further to form stars. Stars are denser than these clouds, so the clouds ended up orbiting around the stars. As the clouds orbited, they formed disks, which then clumped together into planets and asteroids. The solar system seems to be roughly disk shaped, if you look at all the planet orbits:
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