What is the purpose of computing the eigenvalue of a PDE problem?

For your specific question, observe that $$ \nabla^2 u = 0 $$ is a special case of the eigenvalue equation $$ \nabla^2 u = \lambda u $$ with $\lambda = 0$. So understanding the eigenvalue problem certain helps understanding the homogeneous problem.

For the inhomogeneous problem, formally if one knows all the possible eigenvalues $\lambda_k$ and corresponding eigenfunctions $u_k$, then one can try to write $$ f = \sum_{k} c_k u_k $$ as a combination of the eigenfunctions with constant coefficients $c_k$. If this is possible, then we can solve $$ \nabla^2 u = f$$ by formally inverting the operator to get $$ u = \sum_{k} \frac{c_k}{\lambda_k} u_k $$ And so yes, understanding the eigenvalues and eigenfunction can also help understand the inhomogeneous linear problem.

In quantum mechanics, eigenvalues are of physical interest. The eigenvalues of the Hamiltonian operator (which determines how the wavefunction propagates in time) are also the only allowed energies of the system.

When propagating in time, certain physical effects can be simulated by using an incomplete eigenvalue basis. To simulate a "core" electron of a big atom like Xenon, one can make the assumption that the outer electrons remain where they are and leave their energy states out of the core state's basis set to simulate the effect of the exchange interaction on the core. One can do the same with the outer and core electrons switched.

Selection rules allow the bandwidth of eigenvalue bases to be lower than that of finite difference or finite element matrices.

For a classical elastic system, the eigenvalues and eigenstates correspond to the natural frequencies and normal modes of the system. The system will tend to oscillate at these frequencies, so one must take care not to couple the system to vibrations at these frequencies.

Solving a time dependent equation like the heat equation $$ u_t-\nabla^2u=0,\quad x\in\Omega,\quad t>0 $$ with boundary condition $u(x,t)=0$ if $x\in\partial\Omega$, $t>0$, and initial condition $u(x,0)=u_0(x)$, $x\in\Omega$, by separation of variables (that is, looking for solutions of the form $u(x,t)=T(t)X(x)$) leads in a natural way to the eigenvalue problem. The solution can be expanded (under some conditions on $\Omega$ and $U_0$) as $$ u(x,t)=\sum_{n}e^{-\lambda_nt}X_n(t), $$ where $\lambda_n$ are the eigenvalues and $X_n$ the associated eigenfunctions.