The Grassmannian $G(k,V)$ is the set of all $k$-dimensional subspaces of a vector space of dimension $n$. It can be embedded inside $\mathbb{P}(\bigwedge^k V)\cong \mathbb{P}^{\binom{n}{k}-1}$ as a projective variety using the Plucker embedding given by $<v_1,v_2,\cdots,v_k>\ \mapsto\ v_1\wedge v_2\cdots\wedge v_n$.

When $k=2,n=4$, $G(2,4)$ can be embedded inside $\mathbb{P}^5$ and the resulting variety is called the Klein quadric. The image of all 2-dimensional subspaces containing a fixed 1-dimensional subspace is a 2-plane inside the quadric called the $\alpha$-plane and the image of all 2-dimensional subspaces contained in a fixed 3-dimensional subspace is again a 2-plane inside the quadric called the $\beta$-plane. How do we prove this?

Is there a generalization of this to $G(k,n)$? Is the image of all $k$-dimensional subspaces containing a fixed $(k-1)$-dimensional subspace a $(n-k)$-plane inside the Grassmannian variety and similarly is the image of all $k$-dimensional subspaces contained in a fixed $(k+1)$-dimensional subspace a $k$-plane inside the Grassmannian variety? Also what can we say about the images of $k$-dimensional subspaces containing a fixed $r$-dimensional subspace or contained inside a fixed $r$-dimensional subspace? I know that these must be the intersection of the Grassmannian variety with a linear subspace but when are they subspaces contained inside the Grassmannian variety?

Principles of Algebraic Geometry) has a nice introduction to Schubert calculus, and they explicitly work out their general formulas for the case $k=2, n=4$, so this might be a particularly useful resource for the OP. $\endgroup$