It has to do with what happens when a giant nebula of gas and dust starts to form a star. All the gas and dust basically flattens into what's called a "protoplanetary disc" that is rotating in one direction, similar to how water gets sucked down the drain. Moons, planets, asteroids, large bodies, etc. get formed from this disc, so it makes sense that they'd all be in the same plane.

Also explains why most of the objects in the solar system are moving in the same direction as well.

The protoplanetary disk, after going “splat” in one dimension, will continue to contract down as more and more matter gets attracted to the center. But while much of the material gets funnelled inside, a substantial amount of it will wind up in a stable, spinning orbit in this disk.

There’s a physical quantity that has to be conserved: angular momentum, which tells us how much the entire system — gas, dust, star and all — is intrinsically spinning. Because of how angular momentum works overall, and how its shared pretty evenly between the different particles inside, this means that everything in the disk needs to move, roughly, in the same (clockwise or counterclockwise) direction overall. Over time, that disk reaches a stable size and thickness, and then small gravitational instabilities begin to grow those instabilities into planets.

Source article for above.

Further reading:

Rings and disks are common in astronomy. When a cloud collapses, the conservation of angular momentum amplifies any initial tiny spin of the cloud. As the cloud spins faster and faster, it collapses into a disk, which is the maximal balance between gravitational collapse and centrifugal force created by rapid spin. The result is the coplanar planets, the thin disks of spiral galaxies, and the accretion disks around black holes.