Fix your favorite two such subspaces $H$ and $H'$. (Mine are the span of $\{ e_0, e_1, \dots, e_m \}$ and the span of $\{ e_m, e_{m+1}, \dots, e_n \}$ for some choice of ordered basis of your underlying vector space $V$). You want to show that the $PGL_{n+1}$-orbit of this point is the variety of pairs of subspaces that intersect in a point. You know that the orbit is an irreducible subvariety of this irreducible variety, so all you need is for them to have the same dimension. You can compute the dimension of an orbit once you identify the stabilizer of a point.

$PGL(2)$ acts simply transitively on triples of distinct points, so any element of $(\mathbb{P}^1)^4$ can be brought to a unique element of the form $(0, 1, \infty, z)$, where $z$ is any element of $\mathbb{P}^1$ except $0$, $1$, or $\infty$. The book An Invitation to Quantum Cohomology by Koch and Vainsencher has a very good discussion of this moduli space.