Energy is a secondary concept, at least in Newtownian mechanics, so let's start with the fundamentals.
There exists a force field. A gravitational one, say. An object in this field feels a force. This force "wants" to make the object accelerate.
From forces, define work as $force \times distance$ and now we can give precise mathematical meaning to the intuitive concept that "it takes work to lift this ball from the floor to the table".
From work, define energy. Energy is how we keep track of all the work that we did. 131
When we say "the ball has $x$ Joules of kinetic energy" that's a way of saying "some force acted upon the ball, doing $x$ much work, and now because of that force the ball has accelerated to the given velocity".
So let's answer the question: "why does a system like to minimize its total energy?" First we have to clarify the question, we have to nitpick about what "the system" is because when lifting a ball from the floor to the table I have apparently increased its energy, contradicting the premise. I think you mean "why does a system like to minimise its potential energy?" Potential energy exists because the force exists -- the ball potentially will fall back to the floor, if it is allowed to; gravity will do work on the ball as it falls and this will manifest as kinetic energy. So saying "the ball wants to minimise its potential energy" is basically saying "the force of gravity exists".
But if the ball actually is allowed to fall, we're trading potential energy for kinetic energy, and the total energy of the system is constant. If it bounces with perfect elasticity and there is no air resistance etc, it will return to its initial configuration: no energy change. In practice, the bounce will be slightly elastic, work will be done agitating the molecules into motion (heat, sound, etc) and so not all the energy will be returned to the ball in the bounce, so the energy of the (no longer closed) system will tend to decrease. 621