My simple understanding is to keep some basic physical formulas invariant in coordinate system transformation.
For example, it is a formula that many middle school students know. If a particle moves in the x-y plane of the right-handed system, the angular velocity vector along the Z axis is positive. Imagine that the particle and the coordinate system are symmetrical to the other side of the mirror through the mirror. At this time, x'-y'-z' is the left-handed system, and the angular velocity vector is positive along the Z' axis (the angular velocity can be visualized as a directional arrow). But at this time, the speed calculated by the original cross formula is contrary to the actual situation ... In order to make these physical laws not be different because of the transformation of the coordinate system, it is natural to think that everyone should use a set of coordinate system (left or right). Of course, you can also define different calculation rules for different coordinate systems (such as cross multiplication), but isn't that too much trouble?
Therefore, the right-hand coordinate system came into being. As for why they are right-handed, it is irresponsible to guess that most people are right-handed, and the right-handed system looks pleasing to the eye. What is more convenient than gestures?
Ps: The transformation of the left-handed system is problematic because it is a second-order antisymmetric tensor. Who told you to simply write people in vector form!