Your question is an interesting one and I am a little puzzled as to why I myself have not considered this basic question! I had to look into this so I am not 100% sure of the answer but here is my best attempt;
As with all theories we often start with axioms that are simply stated or defined and accepted. All other theories derive from these.
If the sample space has elements, i.e A,B C .. etc where A can contain one or more elementary outcomes then the classical axioms of probability according to Kolmogorov are;
Let A and B be any events then
The definition of conditional probability
satisfies the above axioms.
We can state intuitively that if knowing B does not affect A, i.e , then from the conditional probability definition we can state that
which leads to the multiplication of independent events.
Thus as far as I can see the independent events that multiply comes directly from the definition of conditional probability. If we were to make another definition then this might change.