I have been wondering what's the intuition behind a well known result: $E(XY) = E(X) E(Y) $ for independent random variables $X,Y$

I found this post: [here](https://math.stackexchange.com/questions/1468424/proof-product-of-expectation-of-two-independent-random-variables) which kinda solves the problem.

But, the explanation given there seems to be not clear enough for me.

**What I think:**
Without loss of generality, we know that besides independence we can assume that both random variables, $X$ and $Y$ are simple random variables, and so, it is possible to represent them as, i.e. taking X first:

$X = \sum^n\_{i=1} a\_i 1\_{A\_i}$, then compute the product $XY$ and take expectation. 

But could somebody please explain the intuition behind it to me? 

I really want to get the notion of how to understand the result of that post (which i believe is correct)

Thank you all guys.!