Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle
$$
G\to E\to B,$$
then $B=E/G$, the orbit space under action of $G$. 


Let $BG$ be the classifying space of $G$. 


My question:


How to get the fiber sequence


$G\simeq \Omega BG\to E\to B\to BG$?



fiber sequence of principal bundles
The bundle $E \rightarrow B$ is classified by a map $B \rightarrow BG$, which means that we have a pullback
$$ \require{AMScd}
\begin{CD}
E @>{}>> EG\\
@VVV @VVV \\
B @>{}>> BG
\end{CD}
$$
As $EG$ is contractible, this gives us a fiber sequence $E \rightarrow B \rightarrow BG$. As $E \rightarrow B$ is a $G$-principal bundle, its fiber is $G$ which is homotopy equivalent to $\Omega BG$. This can be seen from the fibrations $G \rightarrow EG \rightarrow BG$ and $\Omega BG \rightarrow PBG \rightarrow BG$ where $PBG$ is contractible. 


