What happens while you sleep
Far from a simple pause, the night is a period of intense internal activity. Several mechanisms essential to your progress happen only, or mainly, while you sleep.
Muscle repair
It's mainly during deep sleep that your body repairs the fibres worked and rebuilds tissue. The release of growth hormone, involved in this repair, is largely concentrated in this phase of the night. Shortening your nights cuts directly into this rebuilding window, and therefore into your ability to get stronger and more muscular.
Energy recharge
Sleep also rebuilds the energy stores used in effort and resets your metabolism. That's partly why, after a good night, a session "feels" much better: the tank is full. Conversely, on short sleep you start already half-empty, and every rep costs more.
The nervous system and hormonal balance
Beyond muscles, it's your whole nervous system that regenerates at night. Coordination, reaction time, strength and motivation all depend on a rested nervous system. Sleep also regulates key hormones tied to stress, appetite and tissue building. Sleeping badly tips this whole balance the wrong way, towards more stress and less repair.
Consolidating learning
Learning a new movement, improving technique on an exercise, automating a skill: all of this also consolidates during sleep. Your brain "files" and reinforces what you worked on. It's one more reason why technical progress, not just physical, depends on the quality of your nights.