Picture a straight line of balls confined to a line with no gravity, friction or anything else. The balls are free to move around the line apart from the fact that every two neighboring balls are connected with an elastic spring which pushes them apart when they get closer and pulls them together when farther than a distance $a$.
When we take one ball on the left end of the line, and move it, it's neighbour on the right will copy it's movement due to the ellastic force between them. Force however only causes acceleration, so the neighbour of the ball we move will always reproduce the original movement only with a slight delay. But once the neighbour of our ball on the left end is disturbed from it's initial position, it also will cause his neighbour on the right side to copy his movement and so the movement is copied throughout the whole line of balls. When you think about it, this is practically a propagating pulse such as a wave and it arises solely from the fact that the spring forces two neighboring balls to copy their movement.
Now why does this converge to a wave equation? We call the axis of the original line $x$, the balls are put at equidistant $x_i,\, i=1,2,3...$ and the spring constant between them is always $k$. The equation of motion for the position $x_i$ is then
$$\frac{d^2 \! x_i}{dt^2} = -k(x_i - x_{i-1}) -k(x_i-x_{i+1}) = k(x_{i+1}-2 x_i+x_{i-1})$$
Where we have accounted for the spring force of both neighbors and the last equality is just adding the two. Now we can investigate the displacement of the ball from it's initial position $\delta(i,t)=x_i(t)-x_i(0)$ and suppose we have so many tiny balls at so tiny distances we can actually just talk about the displacement of the ball at an original position $x$, so we have $\delta=\delta(x,t)$. Then our equation reads
$$\frac{\partial^2 \delta}{\partial t^2} (x,t) = k(\delta(x+a,t)-2 \delta(x,t)+\delta(x-a,t))$$.
$\delta(x+a,t)-2 \delta(x,t)-\delta(x-a,t)$ for $a$ supersmall converges to the second partial spatial derivative times $a^2$ (check this)
$$\delta(x+a,t)-2 \delta(x,t)+\delta(x-a,t) \to a^2 \frac{\partial^2 \delta}{\partial x^2}(x,t)$$
So now we just have a wave equation for the super-tiny-ball-in-a-line displacement.
$$\frac{\partial^2 \delta}{\partial t^2} (x,t) = ka^2 \frac{\partial^2 \delta}{\partial x^2}(x,t)$$
And all this just from a spring-like connection forcing the neighbours to copy each other. I feel it is pretty intuitive that this "pulse copying" or "pulse propagation" will happen also in this continuum limit and also in any processes governed by same equations.
It might seem surprising that the exact shape of the disturbance is propagated, i.e. that the wave-form does not change or decay in any way. In fact, it is just this spring-like "copy" force growing linearly with distance that allows the conservation of the wave-front, and most of the systems have wave equations like this only for small disturbances, i.e. neglecting non-linear forces.
Neglected dissipation also damps the disturbance, but may for example do so quicker for higher frequencies thus dumbing down sharp features of the wave-front quicker. I believe it is a right physical down-to-earth intuition that arbitrary wave-front conservation is just a very approximate concept.
Once non-linear forces become non-negligible and dissipation is introduced, there is no general wave-front conservation, but only special kinds of pulses may propagate without change which hit just the right combination of forces etc. - solitons.
