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?