Why are Lagrangian subspaces in a symplectic vector space interesting? A subspace in a symplectic vector space could be one of two extremes: either symplectic (meaning the form is nondegenerate there) or Lagrangian. Or it could be something between the two, meaning a degenerate nonzero form.
What makes Lagrangian subspaces particularly interesting? For example, we consider the Lagrangian Grassmannian. But why not the symplectic Grassmannian?
 A: The symplecticGrassmanian, as you define it, is just the complement of a codimension 1 subspace of the usual Grassmanian. Hence its geometric properties are very close to the original Grassmanian. However, the Lagrangisn Grassmanian has high codimension, so it’s properties are more new. Also, it is compact, which is useful for many purposes (e.g, making it a flag manifold).
A: If $L$ is a Langrangian in a symplectic vector space $V$, then $V$ is isomorphic to $L\oplus L^*$ with the standard symplectic form. This is a linear analogue of Weinstein's neighborhood theorem.
Recall that, given a vector space $L$ with dual $L^*$, the vector space $L\oplus L^*$ can be endowed with a symplectic form via the formula
$$
\omega_L((v,\xi),(v',\xi'))=\xi(v')-\xi'(v). 
$$
Note that $L$ and $L^*$ are Lagrangian subspaces of $L\oplus L^*$. Now let $L$ be a Lagrangian inside a symplectic vector space $(V,\omega)$. Then $L$ admits a vector space complement $L'$ that is also Lagrangian. This defines a bijection $\varphi:L'\rightarrow L^*$ via the formula $\varphi(v)(w)=\omega(v,w)$. Then the map $\Phi:V=L\oplus L'\rightarrow L\oplus L^*$ with $\Phi(v,v')=v+\varphi(v')$ is a symplectomorphism between $(V,\omega)$ and $(L\oplus L^*,\omega_L)$. 
A: In symplectic linear algebra this is perhaps not completely clear. However, if one passes to symplectic geometry then the Lagrangean submanifolds indeed play a dominant role. This is in some sense Weinstein's Lagrangean creed: Every manifold $M$ is a Lagrangean manifold when viewed as zero section inside its cotangent bundle.
But also beyond this observation Lagrangean submanifolds show up in e.g. Hamilton-Jacobi theory, in completely integrable systems, and in quantization theory where they can be thought of semiclassical limit of quantum states (to some extend). Finally, in semiclassical analysis they turn out to be related to supports of pseudo-differential and Fourier integral operators.
Surprisingly, symplectic submanifolds turn out to be not that important. Instead, coisotropic submanifolds have an important role when it comes to phase space reduction (coisotropic reduction). Also in Poisson geometry, coisotropic submaifolds are in some sense the closest one can get to Lagrangean ones, a notion which no longer makes sense.
I hope this gives some inspiration why Lagrangean submanifolds are useful.
