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In the mathematical sense, I understand what is the solution for a wave equation of such form: $$\frac{\partial^2 u}{\partial t^2}=c^2\left( \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2} \right)$$

But I have a hard time digesting its physical meaning.

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The main feature to focus upon is the principle of superposition of waves i.e. if there are two waves $u_1(x,t)$ and $u_2(x,t)$ then $u_1(x,t) +u_2(x,t)$ is also an admissible solution to the linear wave equation. Thus, physically it is the simple addition (interference) of the amplitudes of the two waves that generates another wave.

For the 1D case

\begin{equation} \frac{\partial^2 u}{\partial t^2}=c^2 \, \frac{\partial^2 u}{\partial x_1^2} \end{equation}

The partial derivatives involved on the L.H.S and the R.H.S in words read as the ratio of the rate of change of the rate of change of amplitude in time to that of space is proportional to the square of the velocity of propagation.

The solution $u(x,t)$ represents the amplitude of the displacement either perpendicular to (in case of transverse wave e.g. vibrating string) or along (Longitudinal wave e.g. sound in a tube) the direction of motion i.e. along the x-axis. At most the wave can travel in either x>0 or along x<0 direction. The speed of propagation of the wave is given by $c$.

When we go to 2D, then the dynamics of a wave-like disturbance emanating from some region (localized) and its spread is not very obvious. For instance, dropping a pebble into still water causes an outward spreading ripples from the source. The same goes for the surface of a drum (bounded region), where the striking act creates undulations of the drum-skin.

$$\frac{\partial^2 u}{\partial t^2}=c^2\left( \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} \right)$$

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In 3D, the wave equation would look like

$$\frac{\partial^2 u}{\partial t^2}=\frac{1}{c^2}\nabla^2 u $$ The solutions are spherical waves in three dimensions. There are many phenomena that utilize the wave equation from geophysical models to general relativity. The nature of the solution i.e. the function $u(x_1,x_2,x_3, \cdots, t)$ will be determined by the kind of physical quantities involved.

For dimensions higher than 3 its difficult to visualize things but an analogy could be made. For instance, a 2D surface wave can be considered embedded in a 3D space. Likewise the 3D wave could be thought of as embedded in a 4D space. The space need not be the real space but of some variables/parameters involved. One thing is certain that all the solutions would involve some oscillating periodic function as a solution, in case the wave equations are linear.

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See:

https://physics.stackexchange.com/a/110842/45664

Paraphrasing from that:

The wave equation can by derived from geometry alone, without using physics. Consider a right moving wave $f(x−ct)$ and consider small changes in $x$ and $t$, ie. $Δx, Δt$ (They each cause a small shift or translation of $f(x−ct)$). Note that $Δx = cΔt$. So $Δf/Δx = Δf/cΔt = (1/c)Δf/Δt$. Doing that again we get

$$\frac{\Delta^2 f}{\Delta^2 x} = \left(\frac{1}{c}\right)^2\frac{\Delta^2 f}{\Delta^2 t}$$

Then letting $Δ$ become very small we get

$$\frac{\partial^2f}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2f}{\partial t^2}=0 $$

From the geometry alone, it was only needed to note that a change in $t$ multiplied by the velocity yields the same results (as measured by the second derivative) as a change in $x$ --that is, a translation of $f(x−ct)$.

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