The important thing here is the the concept of a "phase change" and the fact that phase changes require large amounts of energy. When something turns from solid to a liquid or from a liquid to a gas, it has to take on a great deal of energy to speed the molecules up.
For example, ice water will stay at 32 degrees for a long time as the ice converts from a solid to a liquid at 32 degrees and the ice continues to draw heat from the water to make the phase change as more ice melts. The water in turn draws heat from the glass and surrounding air, and by insulating the ice water and preventing that energy transfer, a styrofoam cup will keep it cooler longer, and slow the melting of the ice compared to the same amount of ice and water in glass or metal cup where more heat transfer can occur.
The boiling point of a liquid also rises with the pressure, so as the pressure drops in a NOS or CO2 tank as gas is used, the temperature at which it boils drops below the ambient temperature and the liquid basically boils and produces more gas. The phase change however cools the tank and liquid so as the temperature of the tank and liquid drops the liquid stops boiling and reduces the flow of gas at any given pressure.
So...you heat a NOS bottle to provide more heat for more rapid phase change of the liquid to a gas and to keep the temperature of the tank and liquid from dropping in order to provide gaeous NOS at a higher rate and more constant pressure. Aluminum is also the metal of choice for a NOS bottle as it transfers heat more efficently than a steel or a composite cylinder.
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Way back in the 1970's they experimented with liquified air diving systems. Basically this involved an AL 80 sized insulated tank of liquid "air" and a heat exhanger to use the heat of the surrounding water to facilitate the phase change from a liquid to a gas.
It had some large technical challenges in terms of the respective boiling points of oxygen and nitrogen, the weight and limited volume of the insulated cylinder, the heat exchanger, the cost of the unit, the need to store cryogenic liquids and the expense of producing them, the liquids boiling off in stroage, etc. In the end the advantages of storing a lot more cu ft of gas as a liquid than as a compressed gas were offset by the cost and complexity of the system.
But I thought the lady modeling the unit was pretty hot.