The reason altitude exposure increases calorie expenditure is energy economics.
At sea level, your body extracts oxygen efficiently. Your cardiovascular and respiratory systems work at a known cost for a given workload. Move that same workload to a hypoxic environment, and the cost of breathing increases. Your heart rate is higher for the same wattage. Ventilation is higher. Recruited muscle has to work with less available oxygen, raising the metabolic cost of force production. Excess post-exercise oxygen consumption (EPOC) is elevated for hours after the session ends.
Burn calories at altitude during training, and the increase is mostly from the cost of the work itself. Sleep at altitude, and the increase is mostly from elevated overnight metabolic demand and the adaptive cost of increasing red blood cell production.
Both effects are real. Both are modest in absolute terms compared to the difference between training and not training. The protocol works as a multiplier on existing effort, not as a substitute for it.