A short exact sequence of groups and their classifying spaces
I think there's a typo in that MO answer: $BG$ is modeled by $EG \times_G E(G/H)$ since this is just $BG \times E(G/H)$ and $E(G/H)$ is contractible.
Now we have a natural map $EG \times_G E(G/H) \rightarrow B(G/H)$ which is a fibration with fiber $(EG)/H \cong BH$.
For a great reference on all this stuff written in a very friendly style see:
http://www.math.washington.edu/~mitchell/Notes/prin.pdf