Phonon-phonon-interaction as higher order terms in the potential Is there a simple way to understand why phonon-phonon-interaction is described by higher order terms in the potential?
I mean: Having a quadratic potential is essential for the definition of the phonons as excitations of an harmonic oscillator. How can you suddenly talk about (interacting) phonons when you don't have a simple quadratic potential anymore...
 A: First off, why are there no phonon-phonon interactions with a quadratic potential? A quadratic potential means Hook's Law; force is proportional to displacement. That leads to a nice linear wave equation. That allows for superposition; waves just add together linearly and don't disturb each other. This all breaks down for non-linear wave equations -- ones that do not come from quadratic potentials.
This also holds for phonons, so it may be helpfull to dig into the above material if you want to know more.

How can you suddenly talk about (interacting) phonons when you don't have a simple quadratic potential anymore...

It's the same reason we say springs following Hooke's law or pendulums with small oscillations are simple harmonic oscillators; neither is true, but they're mostly right and form a good starting point.
We use good, old-fashioned time-dependent perturbation theory. The quadratic potential  $H_0$ is mostly right; the higher order terms $H_p$ (the perturbing hamiltonian) are small compared to the quadratic term. Together:
$$H=H_0+H_h$$
You already have a complete set of solutions for the quadratic potential, so you use them along with the perturbing Hamiltonian and Fermi's Golden Rule. The rate of transition from state $\left|a\right\rangle$ to $\left|b\right\rangle$ is then:
$$R_{b\to a}=\frac{2\pi}{\hbar}\left|\langle a|H_p|b\rangle\right|^2\rho$$
where $\rho$ is the density of states. Those transitions are phonon-phonon scattering.
$H_p$ has other effects too; e.g. it'll change the energy levels a little. However, that's usually neglected because it has much less important than the  phonon-phonon scattering this perturbation brings. You'll need to break out a mid-level quantum book if you want more details.
Why time dependent (rather than independent) perturbation theory? That's a tougher question. If you want to dig into it, it might be useful to start with some of van Hove's articles:
http://dx.doi.org/10.1016/S0031-8914(54)92646-4
http://dx.doi.org/10.1016/S0031-8914(57)92891-4
The short version is that it works, and that it's never wrong to treat things as time dependent, even if they're not.
