You might have already understood the intuition behind it. But here is my view on it.
Conditional Probability: Probability of an event A given that event B happens. That is, Probability of A given B represented as $$P(A|B)= \frac{P(A \cap B)}{P(B)}$$
There lies the subtle idea in the definition of this. Given that event B happened, our sample space got reduced to "number of favourable outcomes for B". Now, we need to find the outcomes in which both A and B occurs given B occurs definitely i.e.,
$$P(A|B)= \frac{\text{Favourable outcomes for the occurrence of both $A$ and $B$}}{\text{Favourable outcomes for $B$}}$$ $$= \frac{\large \frac{\text{Favourable outcomes for the occurrence of both $A$ and $B$}}{\text{All possible outcomes}}}{\large \frac{\text{Favourable outcomes for $B$}}{\text{All possible outcomes}}}$$ $$= \frac{P(A \cap B)}{P(B)}$$
Now, if A and B are independent events, then there is no way B affects A. So, probability of occurrence of A does't change with respect to B, which means
$$P(A|B)=P(A)$$
Combining above two equations we have
$$\frac{P(A \cap B)}{P(B)} = P(A) \Rightarrow P(A \cap B) = P(A)\times P(B) $$