The blog post explained it perfectly. There are 2^32 integers when you pick from [0,2^32). But there are 0x3f800000 floating point numbers between 0 and 1. And the former number is not divisible by the latter number. Therefore using division by 2^32 cannot be unbiased.

It's helpful if you first look at smaller examples. If we were to generate random integers in [0,10) by first generating random integers in [0,50) and then dividing by 5, that's valid. Exactly 5 numbers get mapped to one number each: the numbers [0,5) get mapped to 0, [5,10) get mapped to 1 etc. But what if you do the same division trick if you instead want to get numbers in [0,3)? Do you do the same division trick? Then the probability of the number 2 appearing is less than that of 0 or 1.