State functions are "variables" that define the state of a system. When you have a system you need to be able to define the conditions in which it exists before and after a change. We typically referred to these as the initial and final states. By states we mean the system can be described by a set of properties. For example, the state of a system might be that I have 1 mole of argon in a 10 L container with a temperature of 300 K. Here the state of the system is defined by the "state functions" of volume and temperature as well as the amount of the gas. Likewise, the pressure is also a state function. We'll also see later in thermodynamics that there are a number of variables related to energy that are also state functions.
If I take my 1 mole of gas and do something to it, I might end up in a new state where for example the volume is now 20 L and the temperature is 600 K. For such a process I can look at the change in the state functions. In this case the initial volume, Vi, was 10 L and the final volume, Vf, was 20 L. So my change in volume is given by
\[\Delta V = V_{\rm f} - V_{\rm i} = 20 L - 10 L = 10 L\]
The key to this idea is that the change in volume has nothing to do with the particulars of the mystery process that brought me from my initial state to my final state. The difference in volume will always be the same. That is, if I started with a volume of 10 L and ended with a volume of 20 L the difference is always + 10 L. This might seem frightfully obvious. However, when we start to think about abstract state functions related to energy it can be more difficult to wrap your head around the ideas. But it is important to know that the concept is exactly the same. If you know the initial state and you know the final state, then you can calculate the change (regardless of the process by which you achieved the change).