The physical observation of the conservation of energy? Aside from Noether's Theorem, how do we know energy is conserved?
Energy is the capacity of a system to do work. It's the number that tells me how much "force" a system can apply over a distance. For long I knew we could create forces. However, the only reason... The physical reason that conservation of energy is making sense to me is that when I apply a force to move matter from point $A$ to $B$, I must apply force $x$ on it over displacement $D$ in order to return it I must apply the same amount of force $x$ over $D$. Also the fact that all our natural forces are conservative (in most cases), if work is done by any of the four forces. You must apply the same work again because the natural force is acting against you. Example: Ball is attracted down to Earth's surface by gravity, $9.8$J of work is done ($mgh = 1 \times 9.8 \times 1$) $9.8$N of force has been applied. When the ball is at rest on the surface, I must apply the same amount of force against gravity to move it upwards back to point $A$ ($1$m above the surface). Does this make any sense?
Why is energy conserved besides Noether's Theorem? What other physical observations show it must be true in all our current observable systems? 
 A: Conservation of energy started as an experimental observation. Wikipedia has the history of how scientists of the time arrived to the concept of conservation of energy as a concept.

It was Gottfried Wilhelm Leibniz during 1676–1689 who first attempted a mathematical formulation of the kind of energy which is connected with motion (kinetic energy). Leibniz noticed that in many mechanical systems (of several masses, m_i each with velocity v_i ),



was conserved so long as the masses did not interact. He called this quantity the vis viva or living force of the system. The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction. 



In a paper Über die Natur der Wärme, published in the Zeitschrift für Physik in 1837, Karl Friedrich Mohr gave one of the earliest general statements of the doctrine of the conservation of energy in the words: "besides the 54 known chemical elements there is in the physical world one agent only, and this is called Kraft [energy or work]. It may appear, according to circumstances, as motion, chemical affinity, cohesion, electricity, light and magnetism; and from any one of these forms it can be transformed into any of the others."

It was a long drawn out observational process where many contributed.
The mathematical formulations came and made  empirical observation into a strict law, as explained in the other answer.
