why does a skater spin faster when they pull their arms and legs inwards?

Answer 1

Conservation of momentum.
The energy of a rotating body is proportional to its rotational speed and its diameter.
As the amount of energy is always the same for a given body,when the diameter is reduced the speed must increase.
The same applies to satellites orbiting the Earth. The higher the orbit the slower the satellites speed. This is why a body in geosynchronous orbit must be at a distance of 23,000 miles from Earth.

Answer 2

How, centrete is incorrect or misleading on four counts. (1) The spin-up is due to conservation of *angular* momentum (when one says just "momentum", that generally refers to linear momentum). (2) As the skater's arms are pulled in, she is performing work, and her kinetic energy actually increases. True, energy is always conserved, but chemical energy in muscles is converted to rotational kinetic energy in this case. There is no torque involved, though, so angular momentum is conserved. Wiki angular moment to see how that effects spin rate as the mass is drawn to the axis.

(3) Also, the energy of a given body is most certainly not always conserved, as it can exchange energy with other bodies, though this is not occuring here. (4) Finally, kinetic energy is not proportional to rotation speed and diameter.