The intuition: gravity, spinning (rotation), and friction/drag are the dominant factors which shape the largest scale structures.

Absent rotation gravity wants to pull everything into a ball. If you had a planet that was a cube, the sharp edges would be pulled down to make it round.

If you spin that ball, rotation causes that ball to bulge along its equator. Spin it faster and like a pizza that ball flattens into a disk.

Simplified version of solar system formation:

• There is a large cloud of gas and dust. Gravity slowly pulls it into a rough ball. This ball of gas and dust has some tiny, tiny bit of net rotation, in theory it could have 0 net rotations but the odds of that happening in nature are infinitesimally small.
• Gravity wants to pull the ball into a smaller ball. Friction/drag from the gas and dust bumping into each other slows the particles down, radiating their kinetic energy away as heat.
• Conservation of moment comes into play as the ball collapses. Think of an ice skater doing a trick. When they pull their arms in, they start spinning faster, because the amount of angular momentum must stay the same. The same thing happens to the cloud, as it collapses the average distance of particles from the center goes down ⬇️. To honor conservation of moment, the average rotational velocity must go up ⬆️.
• The increasing rotational velocity causes the ball to bulge along the equator. This new shape increases the rate of collisions which increases the rate of drag which makes it collapse more which increases the angular velocity which causes it to bulge more which repeats the cycle. Over a long time this feedback loop flattens the ball of dust into a disk with a large bulge in the center. (Sort of like what a spiral galaxy looks like).
• This is called the proto-planetary disk. The large budge in the middle will become the star. The material in the disk will begin to undergo similar collapse at a smaller scale to form planets with their moons. What is important here however is that all of the material is rotating in the same direction and physically located either in the bulge or in the disk.
• Eventually the bulge will collapse enough for stellar fusion to light within the newly formed star. The solar wind and radiation pressure from this new star will then blow most of the gas and dust which isn’t part of a planet, moon, asteroid, or comet out of the solar system.
• What is left over from this process is a star and a set of planets all rotating in the same direction in a plane

Now is it possible for planets to escape this plane? Yes but it requires an interaction with something outside the solar system because the disk configuration is naturally stable. We see this effect with regards to moons. Most moons in the solar system orbit along the equatorial planes of this planets. But some don’t. Turns out the location of the planets obits within that plane haven’t always been where they are now and they have mucked with each others moons (and even flipped Uranus on its side).

So why have the planets in our solar system not been disturbed by something outside our solar system? (say another star passing too close) Well, the likelihood of life on earth surviving such a close encounter it basically 0. Our orbit would probably get kicked out of the habitability zone, and we’d be bombarded with asteroids and comets. Close encounters of this type happen fairly frequently near the center of the galaxy where stars are close together and infrequently further out where we are. This has lead some scientists to hypothesize that the “galactic habitability zone” only includes a ring around the edge of our galaxy’s disk.