Perhaps one of the first places where one meets
Lagrangian submanifolds is arise naturally in Hamiltonian Mechanics, because of the classical Arnold-Liouville theoremof Hamiltonian Mechanics. Let me state it here:
Theorem (Arnold-Liouville). Let $(M, \omega, H)$ be an integrable system of dimension $2n$ with integrals of motion $f_1=H$, $f_2, \ldots, f_n$. Let $c \in \mathbb{R}^n$ be a regular value of $f:=(f_1, \ldots, f_n)$. Then the corresponding level $f^{-1}(c)$ is a Lagrangian submanifold of $M$.
Geometrically , this means that, locally around the regular value $c$, the map $f \colon M \to \mathbb{R}^n$ collecting the integrals of motion is a Lagrangian fibration, i.e. it is locally trivial and the fibres are Lagrangian submanifolds.
Furthermore, one also shows that the connected components of $f^{-1}(c)$ are of the form $\mathbb{R}^{n-k} \times \mathbb{T}^k$, where $0 \leq k \leq n$ and $\mathbb{T}^k$ is a $k$-dimensional torus. In particular, every compact component must be a lagrangian torus.
For a proof of this result, see for instance the book by Ana Canas Da Silva "Lectures on symplectic geometry".
