I assume you mean $k$-dimensional planes, not just hyperplanes. Then the space $AG_k(\Bbbk^n)$ deformation retracts onto the usual Grassmannian by sending each $k$-plane $E$ to $rE$ at time $r\in[0,1]$. In particular, these space approximate the classifying space of $Gl(k,\Bbbk)$ as $n\to\infty$.

I assume you mean $k$-dimensional planes, not just hyperplanes. Then the space $AG_k(\Bbbk^n)$ deformation retracts onto the usual Grassmannian by sending each $k$-plane $E$ to $rE$ at time $r\in[0,1]$. More precisely, if $E=a+V$ with $a\in\Bbbk^n$ and $V\subset\Bbbk^n$ a linear $k$-dimensional subspace, then $rE$ denotes the affin plane $ra+V$.

In particular, these space approximate the classifying space of $Gl(k,\Bbbk)$ as $n\to\infty$. Because the deformation retractions above are compatible with the natural inclusions $AG_k(\Bbbk^n)\to AG_k(\Bbbk^{n+1})$, their colimit $AG_k(\Bbbk^\infty):=\lim_\to AG_k(\Bbbk^n)$ still has $\lim_\to G_k(\Bbbk^n)$ as a deformation retract. Therefore, $AG_k(\Bbbk^\infty)$ is a model for $BGl(k,\Bbbk)$.