How does a hurricane finally die down?

A hurricane dies down when it loses its energy source, which is usually warm water at the surface of the ocean.
One of the driving forces of a hurricane is heat energy in oceanic surface waters.
Warm water evaporates more quickly, and warm air rises. As warm, moist air rises, the air gets thinner (less atmospheric pressure), which causes the temperature to drop slightly, which causes some of the moisture to condense as clouds.
That condensation releases heat, which warms it up, which makes it rise, which makes it cool, which makes more condensation, which releases more heat…
So as long as there is warm to feed the hurricane, it is a positive feedback loop, creating stronger and stronger updrafts, which pull more and more air in, generating the hurricane.
If the hurricane wanders northward, it may move into cooler waters, which slow its growth, and eventually reduce its severity. If it moves onto land it loses that warm water source, and so dies down.
The single most important factor in a hurricane losing energy is friction. When the hurricane is over water the friction with the surface of the ocean is minimal. This results in an almost totally unreduced Coriolis effect. The Coriolis effect is the apparent force on wind that prevents air from moving from high pressure to low pressure, and instead causes the air to deflect and the low pressure center does not fill.
When you add friction, which is when the hurricane makes landfall, the wind speed drops and that means that the Coriolis effect is reduced and the winds gradually spiral inward, causing the intense low air pressure at the center of a hurricane to start filling. As the pressure at the center rises the force caused by the pressure differences (pressure gradient force) drops. This causes the wind speed to drop. The whole things becomes a feedback loop and eventually the hurricane will dissipate.
If you ever follow the movement of a hurricane you will notice that without fail the category of the hurricane will drop by 1 almost immediately after it makes landfall. This is also why when a hurricane hits an island the storm often does not dissipate, as it is not over the area of friction for long enough.