Why is intersection of two independent set probability a multiplication process? Why is the probability of intersection of two independent sets $A$ and $B$, a multiplication of their respective probabilities i.e. Why is
$$\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B)?$$

this question is about the intuition behind the definition of independence of sets in a probability space
The fact that the probability of the intersection of independent events $A$ and $B$ is the product of their probabilities is actually the definition of independent events.
before I give the proof---lemme me mention about conditional probability---.(..if you already know skip this para) its represented by 
P(E/F) it means-->Probability of event E given Event F has already occurred. Now if F has occurred the sample space of even E will 'reduce' or get limited to outcomes of F. 
 eg: While tossing a die... 1,2,3,4,5,6 is the sample space. now if event of getting an even number has already occurred then find the probability of getting 2---- since the number on the die is surely an even no: the sample space reduces to 2,4,6 ...so 
P(E/F)=p(E[intersection)F)/P(F) ;
{we take intersection....cuz we need only what is common in both the events ..because event F surely has occurred)

Now here is the proof.... for independent events E and F ....whether event E has occurred or not --it doesn't depend on event F So p(E/F)=p(e[intersn]F)/p(F)=p(E) 
===>>
 p(E[intersc]F)=p(E)*p(F) for independant events

it doesn't simply come from the definition ...see there's the proof!!