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For concreteness imagine the 1-d inhomogeneous wave equation $u_{xx} - u_{tt}=f(x)$, with $c=1$.

I know that $f$ is the forcing term which corresponds to energy being injected (or removed) from the medium, and I understand also how Duhamel's principle uses the fact that $f$ is a force to derive d'Alembert's formula for the inhomogeneous 1-d wave equation.

What I don't get is why the function $f$ is a force. If someone explained the homogeneous wave equation to me, and then asked me about the significance of $f$ in the inhomogeneous equation, it wouldn't occur to me to think of $f$ as a force. Is there an explanation that can educate my intuition?

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    The equation comes from physics, and there the term represents a force. – 2017-02-28
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    Educating one's intuition in the context of PDEs quite often means go and learn physics! – 2017-02-28

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Recall that one way to derive the wave equation is by appealing to Newton's second law. In that setup, there are no external forces acting on the string. You could easily modify this by considering forces that act on the string of the form of $f(x)$. Also, I'm sure you're familiar with ODE's of the form $$\ddot{x}(t)+2\beta\dot{x}(t)+x(t)=g(t)$$ which arise in other mechanics problems. $g(t)$ is interpreted as a forcing function in that context, too. It appears in electromagnetism as well in the form of the polarization field $\textbf{P}$ as $$\nabla^2\textbf{E}-\mu_0\varepsilon_0\frac{\partial^2\textbf{E}}{\partial t^2}=\mu_0\frac{\partial^2\textbf{P}}{\partial t^2}.$$