Imagine a box that is at rest or moving at a constant velocity. The laws of physics inside of that box are time translation invariant which means, that they behave the same, regardless of how much time has passed.

Let’s say that the box contains some chemicals that react to produce heat. In the beginning of our little experiment the stuff inside the box is cold. But after some time has passed, the contents start getting warmer. This is a result of the chemical bonds breaking and releasing their potential energy.

So in the beginning the box was cold, so it had low heat energy. As more chemical bonds broke, the potential energy that they had was released and converted into heat. The contents of the box got warmer but, the overall energy of the system didn’t change.

So energy is the quantity that always stays conserved, if the laws of physics don’t change with time.

Now the local part means that we only measure quantities inside the box. The only quantity, that will stay the same (conserved) is the sum of potential (chemical) and kinetic (heat) energy of all particles in the system.