"Every linear system is the critical point of an objective or energy"...I don't entirely understand what this is saying or why it's true.
It means that every time you're solving a linear system, you're also finding the critical point of an energy. In particular, the energy given above. The reason that's useful is because it gives you an idea for how to start designing algorithms to solve linear systems: start with some initial guess, and "ski downhill"; in fact, that's what we had you do in one of your quizzes.
The reason it's true is that if you differentiate the energy above and set the result equal to zero, you get the original system. (Try it!)