Question 1: I know that $K(\mathbb Z, 2)$ is $\mathbb CP^\infty$ and that $K(\mathbb Z_2, 1)$ is $\mathbb RP^\infty$. But, how about $K(\mathbb Z_2, 2)$? Do we similarly have a good description of this space?
Question 2: Furthermore, the purpose of the above question is to find a fibration $$ Gr(V)^\# \to Gr(V), ~~ \text{with fiber $\mathbb Z \times \mathbb RP^\infty$} $$ which is claimed to be obtained by pulling back some bundle through a map $Gr(V) \to K(\mathbb Z, 1) \times K(\mathbb Z_2, 2)$. And I have trouble in understanding this in details. Thanks!
You don't need an answer to Question 1 to understand Question 2.
If $G$ is any topological group, it has a classifying space $BG$ with the property that homotopy classes of maps $X \to BG$ correspond to principal $G$-bundles on $X$. This correspondence comes explicitly from pulling back the universal principal $G$-bundle $EG \to BG$, and principal $G$-bundles have fiber $G$.
$\mathbb{RP}^{\infty}$, or at least a space homotopy equivalent to it, can be given the structure of a topological group in such a way that $B \mathbb{RP}^{\infty}$ is a $K(\mathbb{Z}_2, 2)$, and also $B \mathbb{Z}$ is a $K(\mathbb{Z}, 1)$. This means that a map $X \to K(\mathbb{Z}, 1) \times K(\mathbb{Z}_2, 2)$ classifies a principal $\mathbb{Z} \times \mathbb{RP}^{\infty}$ bundle, which is in particular a fibration with fiber $\mathbb{Z} \times \mathbb{RP}^{\infty}$.
In general, if $Y$ is any pointed connected space, it can be thought of as the classifying space $B \Omega Y$ of its based loop space $\Omega Y$, which is not a topological group but is good enough so that it still has a classifying space. This means that maps $X \to Y$ can be thought of as classifying principal $\Omega Y$-bundles, and in particular they can be used to construct fibrations with fiber $\Omega Y$, given by taking the homotopy pullback of the universal bundle $E \Omega Y \to B \Omega Y \cong Y$. Also, the based loop space of a $K(A, n)$ is a $K(A, n-1)$.