Ice skater increase of energy

The energy comes from the ice-skater's muscles; they have to work to pull their arms in.

There is no external work done on the skater - the energy is converted from the chemical potential energy stored in the skater's body to kinetic energy.

Work is being done by the force that rearranges distribution of matter. If you want to pull in your arms, you have to fight centrifugal force.You can see it as being in some kind of a force field, much like gravitational, for example.Work done by your muscles while contracting your arms is just integral of a force over some path.