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 $G\_n$.
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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. 



Motivation for classifying vector bundles
At Praphulla Koushik's request I am posting my comments above as an answer, with a little extra detail added. 


Complex line bundles are classified up to isomorphism by their first Chern class. To see this, consider the long exact sequence of cohomology associated to the exponential sequence 


$$0\rightarrow \mathbb{Z}\rightarrow \mathcal{C}\_X\xrightarrow{\exp(2\pi i -)} \mathcal{C}^\*\_X\rightarrow 0$$


where $\mathbb{Z}$ is the constant sheaf on $X$ with values in $ \mathbb{Z}$ and $\mathcal{C}\_X$ and $\mathcal{C}^\*\_X$ are the sheaves of continuous functions and non-vanishing continuous functions on $X$, respectively. Since $\mathcal{C}\_X$ is a fine sheaf, the connecting homomorphism $H^1(X, \mathcal{C}\_X^\*)\rightarrow H^2(X,\mathbb{Z})$ is an isomorphism. Since $H^1(X, \mathcal{C}\_X^\*)$ is in bijection with the set of isomorphism classes of complex line bundles on $X$, we have a bijection


$$\{\text{isomorphism classes of complex line bundles on}\ X\}\rightarrow H^2(X,\mathbb{Z}).$$


Finally, since $H^2(X, \mathbb{Z})$ is in bijection with homotopy classes of maps $X\rightarrow K(\mathbb{Z},2)\simeq \mathbb{CP}^{\infty}$, we obtain that complex line bundles on $X$ are classified by homotopy classes of maps $X\rightarrow \mathbb{CP}^{\infty}$. 


