What is it about the state equation that means it can be fully defined by only two independent variables and not, say, one, or three? Is there an easy way to explain this? Is it just by convention?
Thermodynamics is a study of large systems that we only care about very few attributes. To have only two independent variables, it means you have already specified a lot about the system, like what it is made of and what is its chemical potential (or alternatively, how many particles are present). Then you imagine the system is so large that it samples from its most likely configuration set that is consistent with the total energy. So you are focusing all your attention on total energy, and binning together in your mind all configurations that have that total energy, not caring about any more details of the system beyond what is most likely. However, even when you do that, you find there is still an important ambiguity about how that total energy got into that system, this will matter a lot to the things you care about. So you find there is one specific way that energy can get in there that you can track, and that is the work done on the system by its environment. You find if energy gets in there that way, it has specific important consequences. But energy can also get in there in many other ways, all too complicated to track in detail, but they involve complex interactions that we can all lump together into the concept of heat transfer. So you end up with only two ways you need to track how the energy got in there-- work, or heat transfer. Those two separate ways have different consequences, and are the reason you need two independent parameters to understand what you regard as thermodynamically important about your system.
There is also a wrinkle here, which is that the two independent variables cannot be total work done and total heat added, because you have to track how and when the work was done and the heat added. So you can only track work and heat as infinitesmally the two key parameters, they don't work globally to specify the state of the system. So you need two different global parameters, like temperature and volume, to specify the state of the system. One of them is always extensive (is proportional to the size of the system), because it relates to the amount of energy that is in there, and the other is intensive (does not care about the size of the system), as that constrains how and when the energy got in there (i.e., was it work or heat at each stage of the energy-adding process).
I think you are looking for this. In brief, the number of independent variables would be equal to the number of components plus the number of generalized forces. For the most common case of one component and one generalized force (pressure), you get two independent variables.