Why does moving the decimal point in the divisor and the dividend when dividing by a decimal work?

Moving the decimal point one place to the right is the same as multiplying both, he numerator and the denominator by 10.

That is, you are effectively multiplying by 10/10 which equals 1. And since multiplication by 1 does not change the result, it is OK to do that. Similarly, moving two palces to the right is multiplying by 100/100. Similarly moving left.