Locations in Newtonian space can be described as 3 coordinates, or a 3 dimensional space. So can forces and velocities and the like.
As it turns out, the ways things can spin can also be described by 3 dimensions.
Imagine you have something that is spinning. First, draw a unit sphere around it, and where the axis of spinning crosses the sphere record those points. Those two points are on opposite sides of the sphere.
This doesn't fully describe the spin, but tells you where the axis is.
We then need to add on how fast it is spinning, and what direction.
For how fast, we can scale up the distance from the origin. This is nice, because "no spinning at all" has a distance of 0, and that corresponds to the point where the "axis" is undefined. Pretty perfect really that they cancel out!
The next problem is what direction the thing is spinning in. Right now we have 2 points on the sphere and two directions we need to distinguish between.
The right hand rule is literally saying "Ok, we'll pick one of these two points to distinguish spinning one way from the other". 432
If you selected the other point, all of the math would still work, you'd just have a left hand rule, and some functions for converting between angular vs linear math would have a sign change sometimes.
The point we assign to a spin isn't a velocity or an offset. It is just something that behaves mathematically like an offset.
The natural mapping between the space off offsets and the space of rotations -- or, the space of vectors and bi vectors in 3 dimensions -- has a choice to make -- there are two equally good mappings with nice properties. Our choice of which mapping we use is the right hand rule.
To map this over to something you might be more used to, where is the zero point of gravitational potential energy? Sometimes you end up putting it at infinity, other times at ground level. Where you put it doesn't matter; but it does matter you do your math while agreeing on where it is.
You can avoid having to make that choice by only talking about differences in gravitational potential energy, but that sometimes makes the math a tiny bit more annoying to use. So often you end up picking some height or point and saying "all of my gravitational potential energy values are relative to this point".
That choice -- of where 0 is -- is free, but has to be agreed upon in all the math. The same is true of right hand vs left hand rule -- the choice is free, but it has to be the same or the math gets messed up.
We could avoid making the choice, but then it makes other math more annoying to do. So we pick one and stick with it. 499