We see images of Tokamak plasma with all sorts of colours from red to purple. Why do we see any light at all, since the plasma should be so hot to have dissociated all its electrons? It is all from contamination or unwanted cooling?
The sun is mostly plasma, and it almost fits well the black body radiation curve, which is the way we gauge its average temperature: almost 6000K which radiates away. The center is much hotter ( see below).
It is reported that the ITER plasma will be ten times hotter than the sun at the center. This will squeeze the black body type curve towards high frequencies, but there will still be a tail in the visible at the outer cooler layers, to give this type of visible radiation:
In the depths of the Sun where fusion reactions produce the energy that we perceive as light and heat, temperatures reach 15 million °C. In the centre of the ITER plasma temperatures will soar to between 150 and 300 million °C.
In the heart of the Korean tokamak KSTAR, in operation since 2008, a plasma pulse burns brightly. But don't be fooled—the brightest areas of the photo are in fact the coolest. At 150 million °C (the temperature in the centre), the plasma doesn't emit in the spectrum of visible light. © National Fusion Research Institute, Korea.
So the black body curve is really squeezed to the left.
In general from a temperature and over there will be radiation in the visible from black body radiation, but for a plasma there are differences..
The theory of black-body radiation in thermal equilibrium with a homogeneous and isotropic plasma is presented. The relevant thermodynamic quantities of the radiation are obtained. The presence of the plasma changes qualitatively and often quantitatively the concept of black-body radiation, essentially through the effect of plasma density.
In most cases it's a fake: they often use test gasses with the explicit goal of being partially ionized so they give off light and thus allows us to take fancy photographs. The ultimate example of this was the DCX at the 1958 Atoms for Peace where they injected xenon (IIRC) to make a visible glowing ring. Its also possible to take pictures during startup, when it isn't entirely thermalized. There are some youtube videos showing the MAST startup and you can see this happen.
As Maury Markowitz mentioned, the color of the edge of the plasma (the core is colorless and transparent) is determined by the composition of the partially-ionized gas that is recombining with plasma electrons and undergoing transitions at the edge of the plasma. Sometimes this gas is purposely injected (as was the case with DCX and xenon gas), and sometimes it is simply residual gas in the chamber. For the tokamak MAST, it is the latter; since the plasma is made of deuterium, the optical radiation consists of the spectral lines of the dueterium atom, which gives a pinkish glow to the edges of the plasma.
This is consistent with the explanation given by the ITER tokamak's public pages here: https://www.iter.org/newsline/258/1512
As you correctly state, a plasma is composed by a certain density of charged particles (ions, electrons,...). Due to many different reasons, such as the presence of external and internal (self consistent) electromagnetic fields, these charged particles are moving under the action of various forces. It is possible to show that a charged particle that is accelerated will be emitting electromagnetic radiation (not only in the visible spectrum).
Emission of electromagnetic radiation due to the accelerated motion of charged particles in a plasma is often an important part in the energy balance of a plasma system (as the Tokamak configuration you are talking about).