Here is a trivial example that I read from a survey article written by Arnold in the late 80s.
Consider $T^*S^1$, the cotangent bundle of $S^1$ which we can identify with the product $\newcommand{\bR}{\mathbb{R}}$ $S^1\times\bR$. I will denote the obvious coordinates on this cylinder by $(\theta, t)$.
Like any cotangent bundle, $T^*S^1$ carries a symplectic structure, and in this case, any curve on this symplectic manifold is a lagrangian submanifold. However, there are curves, and there are curves.
Take for example the curves $C_\tau:=\lbrace t=\tau\rbrace$, $\tau$ a nonzero constant, which are disjoint from the zero section and are deformations of the zero section via the symplectic flow
$$ (\theta,t)\mapsto \Phi_\tau(\theta,t)=(\theta,t+\tau). $$
Consider next a smooth function
$$ S^1\ni\theta\mapsto f(\theta). $$
Its differential is a section of $T^*S^1$, and its graph $\Gamma_{df}=(\theta,f'(\theta))$ intersects the zero section along the critical points of $f$.
The lagrangian $\Gamma_{df}$ is a rather special deformation of the zero section: it is a Hamiltonian deformation, the points of intersection of $\Gamma_{df}|$ correspond to the periodic orbits of the Hamiltonian deformation.
Why is this fascinating? Certain pairs of lagrangian subspaces intersect in more points than predicted by topology alone, which is in itself an indication that symplectic topology is a bit more rigid than smooth topology alone.
How does the above trivial example fit the general picture?
A lagrangian submanifold $L$ of a symplectic manifold has a tubular neighborhood symplectomorphic to $T^* L$. Thus the case of cotangent bundles can be viewed as local situations of the more general cases. of lagrangian submanifolds and their hamiltonian perturbations.
Given a Hamiltonian flow $\Phi_t$ on a symplectic manifold $X$, the graph of the time $1$-map is a lagrangian submanifold in $X\times X$. Its fixed points correspond to the intersection of the graph with the diagonal in $X\times X$, which is another lagrangian submanifold. Thus the problem of intersection of lagrangian submanifolds contains as a special case the problem of existence of periodic solutions of hamiltonian systems.
Leaving aside the mysterious rigidity of symplectic topology alluded to above, the problem of existence of periodic orbits of hamiltonian systems has fascinated many classics, such as Poincare, because of it's obvious connection to the many body problem and the philosophical question: does the history of our planetary system repeat itself?