The statement I am familiar with regarding classification of vector bundles is :
Given a paracompact space $X$. The set of isomorphism classes of rank $n$ vector bundles over $X$ is in bijective correspondence with the set $[X,G_n]$ of homotopy classes of maps from $X$ to $G_n$.
I am more or less comfortable with the proof of this result.
I could not guess how some one who has done this for the first time thought about it. Were there some spaces $X$ where it is immediately visible that the vector bundles over $X$ have some relation with maps from $X$ to $G_n$?
Question : How did some one guess about the possibility of vector bundles over $X$ being related to homotopy class of maps $X\rightarrow G_n$?
Pointing out a paper where this result is published definitely be useful if it contains some motivation how did the author(s) thought about this.