I am having some trouble understanding aspects of the proof that shows that a numerable $G$-principal bundle $E\rightarrow B$ is a universal $G$-bundle if $E$ is contractible. The proof starts off by claiming that the associated fiber bundle $E\times_G EG \rightarrow B$ is shrinkable (which I think means that it is isomorphic to the trivial bundle). From there one deduces that $E\simeq EG$.
This is what I am having trouble with. Could someone please explain to me why the contractibility of $EG$ already implies triviality of $E\times_G EG \rightarrow B$?
The (equivalent) notion of universality I am aiming to show is that, given any principal $G$-bundle $p:E\rightarrow B$ there is (up to $G$-homotopy) a unique $G$-map $E\rightarrow EG$.
Any answer is greatly appreciated!