Working on $L^2([0,L])$, the operator $A = d^2/dx^2$ (with Dirichlet boundary conditions) is Hermitian. Diagonalizing $A$ is equivalent to solving the ordinary differential equation $$\frac{d^2f(x)}{dx^2} = \lambda f(x)$$ with boundary conditions $f(0)=f(L)=0$. But we already know what the solutions to this equation are. They are hyperbolic cosines and hyperbolic sines if $\lambda >0$, but none of those match the boundary conditions. They are ordinary cosines and sines if $\lambda =0$, and those do match the boundary conditions for certain values of $\lambda$. Those are the eigenvalues of $A$. The upshot, as explained in this video, are that the eigenvalues are $-n^2\pi^2/L^2$, and that the eigenvectors are the functions $\xi_n(x) = \sin(n\pi x/L)$. In other words, Fourier series is the same thing as decomposing in a basis of eigenvectors. [Note: when working in a vector space of functions, we sometimes call the eigenvectors eigenfunctions, but they're still vectors in the abstract sense: elements of the vector space $V=L^2([0,L])$.]

There are three natural ways to think about a function $f(x,t)$, and about partial differential equations like the wave equation $$\frac{\partial^2 f}{\partial t^2} = \frac{\partial^2 f}{\partial x^2}.$$

1. $f(x,t)$ is a number that depends on $x$ and $t$. The wave equation then relates how $f$ changes with $x$ to how $f$ changes with $t$.
2. The dynamic picture: $f(x,t)$ is a function of $x$ that evolves in time. We think of ${\bf f}(t)$ as living in some space of functions, like $L^2([0,L])$, and the wave equation relates the evolution of ${\bf f}$ to an operator acting on ${\bf f}$. In other words: $$ \frac{d^2 {\bf f}}{dt^2} = A {\bf f}.$$ We can then solve the equation by diagonalizing $A$. (Next section.)
3. The static picture. ${\bf f}$ is a function of $x$ and $t$, and doesn't change at all. It lives in some vector space $W$ of functions of two variables. Solution to the wave equation are elements of the kernel of the operator $B= \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2}: W \to W$. Since kernels are subspaces, linear combinations of solutions are solutions. This is called the superposition principle. Ultimately the goal is to find a basis for $ker(B)$.