You have to cool down first

Sleep begins on the falling edge of your temperature curve.

Body temperature is not constant. It runs on a daily cycle, peaking in the early evening and falling through the night to a trough in the small hours. Sleep onset is locked to the falling part of that curve, and the drop is around one degree Celsius.

The drop is not achieved by making less heat. It is achieved by dumping it: the blood vessels in the hands and feet open and radiate warmth away. This is why cold feet make it hard to fall asleep, and why warm feet make it easy — warm extremities are what a body successfully losing heat looks like from outside.

A warm bath helps by making you cooler, not warmer.

That also resolves the bath paradox. In studies, a warm bath an hour or two before bed went with falling asleep sooner, which sounds backwards until you notice what happens when you get out: the heat drawn to the surface leaves quickly, and core temperature falls further and faster than it otherwise would.

A cool room does the same thing more cheaply. Most people do best somewhere around 16 to 19°C, which is cooler than most bedrooms actually are. Of everything on this shelf, it has the best ratio of effect to effort.

The longer answer

What happens when you fall asleep follows the temperature drop into the first minutes of sleep, alongside the heart and the breath. A hot shower before bed puts numbers on the bath advice — in a 2019 review of 17 studies, water of 40 to 42.5 °C one to two hours before bedtime went with falling asleep sooner — and does a cooling mattress help? asks what the bed itself can add.