Still from animation explaining State function in Physical chemistry (click to enlarge)

State variables, or state functions, are one of the most interesting concepts in science. They have a special place in thermodynamics because of their fundamental property of being unaffected by the way chosen by the system to reach its final state.
A good example, to understand how this works, is to consider the distance between Rome and Montreal. This distance is fixed; however, to travel from Italy to Canada you have to choose a path: maybe you’ll choose a straight Atlantic route; or maybe you’ll prefer the opposite direction, to visit Africa, India, Japan, Australia, the Pacific Ocean, and (finally!) reach Canada. In either case, the distance between the two cities does NOT change: what changes is the path you choose.
What does this mean? To keep things simple, we have discovered that some entities (like the absolute distance) depend only on two variables, initial state and final state; while others (like the travelled distance) depend on the chosen path. Like during your holiday journey.
The reason for scientists to like state functions is clear, I believe: you can choose to do what you want, to your specimen. You can follow any path you can (or want), and at the end of all you’ll be assured that the system has changed some properties in a way which depends ONLY by the initial and final states!
The applications are endless. As a chemist, my favorite one is derived Hess’s law, which I paraphrase; if you have no idea about a reaction, just combine other reaction until they sum up to your desired one. When you have done so, the enthalpy of your reaction (which translates into “heat produced or consumed by the reaction”) is the sum of the enthalpies of the known reactions you have used.
This is a powerful tool indeed! For example, engineers use this law to foresee the potential use of undiscovered reactions to power engines, and chemists to predict whether to add or subtract heat from their new reactions to raise the relative yield.
I believe that this is the reason why in the Physical chemistry textbook, this topic extends for two full pages in Chapter 2, as well as three animations! I do however think the the animation regarding the ocean as a reservoir belonged elsewhere (like with the first law of thermodynamics). However, the animation has a connection with the previous one (regarding temperature changes), so I admit that the positioning makes some sense!
One cannot underestimate the importance of clear examples. The ones in these pages are really useful and well-presented.