Answer to this question should start from why we want the physical observables to be represented by linear operators.
Theoretical physics is about constructing a mathematical model which we hope describes the phenomena it's being modelled for and hence helps predicting stuff. In classical physics this mathematical model is based simply on the real numbers (Atleast locally) because of the nice behaviour of things. In quantum mechanics it's not the case. Experiments started giving discrete values as well as continuoous values(such as energy of electrons,etc.). So there is a need for some class of mathematical objects that give equal importance to continuous and discrete cases.
We know linear operators have the property that they posses both discrete and continuous spectra which can act as the required class of mathematical objects. Hence we start identifying physical observables with appropriate linear operators.
Postulate 1. To each dynamic variable there exists a linear operator such that possible values are the eigenvalues of the operator.
We need some place where all the physics happens and where these operators act to give us the required results. So we construct a hilbert space consisting of states of the system which we are observing.
In quantum mechanics the measuring process plays an important role. It will alter the state of the system it's supposed to measure. If we are going to perform two experiments one after other then there is a possibility that some of the information is changed. Suppose $A$ corresponds to a dynamic observable A and $B$ corresponds to the dynamic variable B. Then the product $AB$ corresponds to measuring the observable A after measuring B. If the measuring process is going to alter the result of the next experiment such that, measuring A after measuring B gives different values as measuring B after measuring A then we say they don't commute. Hence it means the commutator is not equal to zero. Otherwise it's zero.
The commutator tells how the measurements are going to alter things.
