Why is recombination rate in an intrinsic semiconductor proportional to product of number of mobile electrons and number of holes?
Let's look at this from the perspective of the electron:

Assume you have an electron at some point in a semiconductor. Now, you also need a hole in the vicinity. The probability of a hole is proportional to the overall number of holes.

Now, taking a step back, you'll need an electron first – and the probability of having a hole at a certain point is again proportional to the overall number of electrons.

So, it needs "luck" twice, getting an electron in a certain point, then getting a hole in the same. (the order doesn't actually matter – you need both.)

So, make an analogous experiment: Flip a coin (probability of say head=1/2) to figure out whether you get an electron, and roll a die to figure out whether you get a hole (probability of a say ⚅ = 1/6). What's the probability that you get an electron (head) and a hole (⚅) at the same time? 

It's the product of these two probabilities, because both events are independent from each other.