There are several energy systems that are at play here.
You have your quick stores within your cells which can be used immediately, no oxygen required. And then you have your energy system which consumes energy without the presence of oxygen. Because there is a higher demand for energy in such a short period of time, your body forgoes using oxygen to produce energy and produces it without it. The by-product is lactate and hydrogen ions (lactic acid). That leads to the burning feeling you get when you hit the sled and often deters the power output through out the movement. Often that's associated with fatigue. You become out of breath because the body needs time to catch up metabolically (buffering of hydrogen ions, converting lactate back into usable energy, normalizing your oxygen levels, and etc..)
Another interesting theory is calcium depletion within the cells of the muscles. It's literally "squeezed" out and the muscle can no longer relax and tense fluidly because of calcium's role in muscle contraction. That's the feeling you get when you're pushing harder and harder, yet the sled isn't moving any faster or your opponent seems to weigh 40 pounds heavier than usual.
Over time the body adapts and is much quicker to recovery after each bout. It has good carry over into endurance based activities such as running or basketball since lactic acid is being cleared out much more quickly, and the body is able to handle being in an oxygen depleted state much longer.
Deeper level detail to play off of marmel75's explanation.
