Why does BH method for False Discovery Rate work?

I want to know why Benjamini Method method for False Discovery Rate works. The critical values for the ordered p-values seems to be (i)*alpha/m for each of the i-th ordered p-value of m p-values. The smallest critical value is then 1*alpha/m. The next is 2*alpha/m, up to m*alpha/m for the largest p value. My understanding is that if Pr(i)<(i)*alpha/m then that indicates a significant (positive) result. I appreciate that there are some tweaks to this basic method. So, why does this work?