I have a question about the explanantion of the idea behind the classiyfing spaces $BG$ with respect (topological) group $G$. In wikipedia is stated that
The classifying property required of $BG$ in fact relates to $\pi: EG \to BG$. We must be able to say that given any principal $G$-bundle
$$ \gamma: Y \to Z $$
over a (nice enough) space $Z$, there is a classifying map $\phi$ from $Z$ to $BG$, such that $\gamma$ is the pullback of $\pi$ along $\phi$. In less abstract terms, the construction of $\gamma$ by 'twisting' should be reducible via $\phi$ to the twisting already expressed by the construction of $\pi$.
Question: In this less formal explanation is used the word 'twisting'. What does it mean to say informaly in context of topology or more concretely the theory of covering spaces to construct a cover or principal $G$-bundle by a 'twisting'? What is here this 'twisting' procedure concretely? Does it refer to cocycles $U_i \cap U_j \to G$ which 'encode' $Y$ and $Z= \bigcup_i U_i$? Or is it something else?