>Why is it happening so? Sounds way to simple to be true in something so complicated as space. Even electrons have 3-dimensional orbits but planets somehow don’t?

Astronomer here:

I would like to clarify that, while each planetary system is on a plane, and the galaxy is on a plane, they are not all the same plane. The planes of the planetary systems are essentially random, and not aligned with the galaxy. Our own solar system is not aligned with the galaxy either.

The reason why spiral galaxies and planetary systems come out in planes has to do with the fact that they are all born from a gas cloud. Any initial gas cloud has some initial angular momentum. As it collapses by its own self-gravity, it has to spin faster to conserve angular momentum. This by itself is not enough to make the gas form a disk. The last ingredient is that gas in space actually behaves like gas ---- it feels pressure, it emits energy. So the initial gas cloud is a "blob" with a lot of random motions, pressure (think gas drag) dissipates most of the random motions, leaving only the "average" motion, which would be a rotation in some direction corresponding to the net initial angular momentum of the blob.

Stars in the galaxy are on a plane because stars from form the gas, and the gas was on a plane. Planets are on a plane, because planets form from the gas (well, Earth forms from the 1% of dust inside the gas) and the gas is on a plane.