Replacing the classical variables simple by the operators in quantum mechanics

While defining the concept of angular momentum in quantum mechanics, I see that in definition they have replaced the classical variable by their respective operators. I want to know what’s the logic behind doing this. Also, I am aware that doing this leads us to some very important results such as magnetic quantum numbers and so on which have factually proven the theory but I am still interested in knowing that how is it correct to just replace the classical variables with the operators.

Also, I am not afraid to hear if I have asked something fundamentally wrong as I am still learning and a novice so please do tell me if that is the case.