Direct limit of $CW$ complex and infinite Stiefel manifold

Question

Let $V_{n}(\mathbb{R}^k)$ be the Stiefel manifold of ortogonal $n$-frames in $\mathbb{R}^k$ and $G$ a compact Lie group. A classifyng space for group $G$ is a connected topological space $BG$, together a principal $G$-bundle $EG \rightarrow BG$ such that the following is true. For any compact T2 space $X$ there is a one-to-one corrispondence between the equivalence classes of principal $G$-bundles on $X$ and the homotopy classes of maps from $X$ to $BG$. So why this definition implies that if we have a principal $G$-bundle $E \rightarrow B$ with the property that the total space $E$ is conctratible, then $(B,E)$ is a classifyng space for $G$? If we consider the $GL(n)$-principal bundle $V_{n}(\mathbb{R}^k) \rightarrow Gr_{n}(\mathbb{R}^k)$ we have to consider the direct limit on $k$ of $V_{n}(\mathbb{R}^k)$ obtaining $$ \underrightarrow{\lim}_kV_n(\mathbb{R}^k)=V_n(\mathbb{R}^\infty) $$ but what is really the direct limit? How can I imagine it and write it formally?