Why is [O III] a good density probe in interstellar medium? According to Draine in his book "Physics of the interstellar and intergalactic medium" (page 210/211) and Caltech [O III] line ratio's (amongst others) are a good density probe, but I cannot really understand why that is the case. Can someone explain me how this works? It doesn't have to be numerical or with detailed formulas, just the idea what is going on.
The key to understanding this is the concept of forbidden lines and forbidden transitions.

A forbidden transition is one that cannot occur radiatively via an electric dipole interaction. Instead, it must proceed either through magnetic dipole or electric quadrupole emission, with much lower probability, or the transition may be accomplished through collisional (de)excitation. 

A forbidden line is the radiation at a wavelength corresponding to a forbidden transition. In laboratory plasmas these are usually not seen - hence the term forbidden lines - because the transitions are normally accomplished through collisional de-excitation on a timescale much shorter than the radiative lifetime through magnetic dipole/electric quadrupole emission. However, in astrophysical plasmas the densities can be many orders of magnitude lower than even the best laboratory vacuums. In this case, forbidden lines are seen and indeed are often an important means of radiative cooling.

The property of forbidden lines which makes them an excellent density-sensitive tool is that they can be "quenched" if the density becomes large enough to make collisional de-excitation more probable than radiative de-excitation. The strength of the forbidden line is thus sensitive to the electron density (which is what dominates the collisions) between densities where "quenching" starts to become effective and higher densities where the line essentially becomes unobservable.

Taking the specific example of OIII (doubly ionised oxygen) optical line emission. There are three forbidden transitions of interest between the $^1D_2 \rightarrow\ ^3P_2$ (501 nm), $^1D_2 \rightarrow\ ^3P_1$ (496 nm) and $^1S_0 \rightarrow\ ^1D_2$ (436 nm) states.
These transitions produce optical forbidden lines which are quenched at different characteristic densities. Measuring a line strength ratio is important because the ratio will be independent of the abundance of OIII. The ratio will depend on electron density, providing the electron density is $&gt;10^{5}\ cm^{-3}$, and temperature (at lower densities, the ratio is just temperature dependent).

To get the density in this case requires more information - usually supplied by a similar line ratio for something like NII, which has a similar set of forbidden lines, but with differing density and temperature dependence.

A more straightforward route is to use the line ratio for forbidden lines of OII or SII where there are a closely spaced pair of energy levels, both undergoing forbidden transitions to the same lower level. In this case the ratios are density sensitive, but not temperature sensitive, again over a range of densities where one or other of the transitions is in the process of being quenched (e.g. for OII 372.6/372.8nm, $10^2 &lt;n_e &lt; 10^{6}\ cm^{-3}$.)