The details do depend on opacity, but the basic phenomenon can be understood without reference to what is happening with the opacity (as explained, for example, in Kippenhahn and Wiegert, Stellar Structure and Evolution, pg. 75). The main point is, if you consider the interior structure, it supports a radiative diffusion rate that determines a luminosity that the envelope of the star will simply have to handle somehow. A main-sequence star has nothing drastically unusual happening to its envelope, so you cannot really alter the stellar radius much. Given the luminosity and the radius, the surface temperature is more or less handed to the envelope by the Stefan-Boltzmann law, so the envelope just has to deal with that somehow. When the surface temperature handed to the envelope is large, there's no problem, but when it's smaller, there is a problem as we shall see.
Now, if you assume the stellar envelope transports heat predominantly by radiative diffusion, the imposed temperature structure encounters no particular difficulty if the surface temperature is allowed to be rather high (say above about 10,000 K when you put in the details of the opacity). Above that temperature, it turns out that the structure of the envelope is pretty insensitive to what that temperature is, and the envelope happily diffuses out whatever luminosity is required because radiative energy diffuses easily when the temperature is high. (This is a consequence of the fact that the energy density stays high if the temperature stays high, so you don't need much in the way of a diffusion speed to get out the luminosity.) However, if your requirement is that the surface temperature be well below 10,000 K, then you have a serious problem, because at those low temperatures, radiative energy does not diffuse easily-- it requires a high diffusion speed because the energy density, which scales like T^4, is so low. In fact, you need a steeper temperature gradient than is stable to convection in order to get the luminosity out. So the star finds a different mode for transporting the heat, it goes convectively unstable and moves hot parcels of gas upward instead of diffusing radiation. This also reduces the temperature gradient to something that keeps the temperature from going to zero before you get to the surface of the star (a problem that radiative diffusion has when it becomes inefficient at lower T).