How does lowering your thermostat save energy? Say it is winter and the outside temperature is 0 degrees f.
I set my thermostat to 74 degrees.  When the temperature inside my home reaches 72 degrees, the furnace will kick on and warm the house to 74.
Now, if I set my thermostat to 68 degrees to save energy, when my house reaches 66 degrees, the furnace will kick on and warm the house to 68.
In both instances, the same amount of energy was expended to raise the temperature by 2 degrees.
So how is it that keeping your thermostat at 68 will save energy?
 A: Forget about the furnace kicking in at 2 degrees below the preset or whatever. This is immaterial. The heat is not needed to produce this temperature rise, but rather to compensate heat loss with the surroundings, and thus maintain the temperature.
You can clearly see that higher temperature presets need a higher amount of heat to be maintained, because the higher the temperature difference with the outside, the larger the heat dissipation with the surroundings, and that is the heat the furnace will have to produce.
If you intuitively think maintaining some temperature requires the same amount of heat per unit time regardless of temperature value, just think how much heat you would need to maintain the same temperature as the surroundings (zero), then to maintain some higher temperature (more than zero).
A: The rate at which your home loses heat is proportional to the difference of temperature between the inside and the outside. This is Newton's law of cooling. Hence, a higher temperature home will lose heat faster.
This means that if your thermostat is set lower, the furnace will need to turn on less often.
A: Firstly, the specific heat (more precisely the specific enthalpy) of humid air varies a bit with temperature, so going from 72 to 74 degrees F will take a bit more energy than going from 66 to 68 F.
However this effect is not very large compared to the effect of heat lost through the house's walls, floor, and roof to the colder outside environment.
To a good approximation, the heat lost through conduction is proportional to the change in temperature between inside and outside. The constant of proportionality depends on the roof/wall/floor material and any insulation.
Say it's 40 degrees F outside. The rate of heat loss will then be greater at an interior temperature of 72 F than at 66 F due to the larger change in temperature.
There is also heat loss due to radiation from the windows as well.
There is a NIST article at this link that covers this in much greater detail.
A: Even ignoring losses, the higher temperature will be more expensive because of the cost of getting to 72% in the first place.
