A skater speeds up by pulling her arms in

Conservation of angular momentum, demonstrated without any equipment.

A skater speeds up by pulling her arms in because doing so brings her body's mass closer to the axis she is spinning around, and the total spin she carries, her angular momentum, stays the same throughout the move, so bringing the mass in closer forces the remaining spin to happen faster to keep that total unchanged.

The physics of a skater's spin

Angular momentum is a fixed quantity for a spinning body as long as nothing outside is pushing or twisting on it, and it combines how fast something is turning with how far its mass is spread out from the axis it turns around. A skater spinning with her arms stretched wide has a good deal of her body's mass sitting well away from the vertical axis running through her, and pulling those arms in close to her body moves that mass much nearer to the axis, sharply reducing how far it sits from the centre. Because the total angular momentum has nowhere to go, nothing outside her is adding or removing any, the only way to keep the total the same after the mass has moved inward is for the spin itself to speed up, and it does, often dramatically, the instant the arms come in. Extending the arms back out reverses the whole effect just as cleanly, slowing the spin down again without the skater needing to do anything except redistribute the mass she already had the whole time. No extra push is added at any point during the move, and none is needed, since the entire change in spin speed comes from redistributing mass that was there from the start, not from any new force applied to the body once the spin is already under way.

The office-chair comparison

Sitting in a chair that spins freely, giving it a firm push with the arms held out wide, and then pulling the arms in tight against the chest reproduces the same speeding-up on the spot, without skates, ice, or any training at all. The chair noticeably picks up speed the instant the arms come in, and pushing them back out slows the spin down again just as clearly, because the same rule applies to a body in an office chair as to a skater on ice: the same total spin has to be carried by whatever mass is spread around the axis, and moving that mass inward means less of it needs to move as far, so what remains has to turn faster to keep the total the same. Trying the same test with a friend giving the initial push tends to make the effect even more obvious, since a stronger starting spin gives more speed for the arms coming in to amplify, and the sudden lurch forward as the arms tuck in is often enough on its own to convince anyone who doubted it.

The number that matters here

Pulling the arms in by roughly half the distance they were held out can noticeably increase spin speed, since the effect depends on how the mass is distributed rather than on speed directly, and skaters exploit this by starting a spin with arms and even a leg extended as wide as possible, maximising how much speed there is to gain once everything is drawn in tight against the body. A skater who starts the spin with her arms already close to her sides has far less mass to bring in and therefore far less speed to gain from bringing it in, which is why the move is always shown starting from as wide a position as the body allows.

Where this stops being true

The speeding-up stops the moment friction between the skates and the ice, or the chair's own bearing, finally bleeds off the spin entirely, since conservation of angular momentum only holds while nothing outside the spinning body is acting on it, and both friction and the deliberate drag of a skate blade dug into the ice are exactly the kind of outside interference the rule assumes is absent. This is also how a skater ends the spin on command rather than needing to wait for it to run down on its own, extending the arms to slow it and then deliberately digging a blade edge into the ice to bring the remaining spin to a stop. An office chair left to spin down on its own, by contrast, keeps the rule intact for far longer, since a good bearing removes so little angular momentum per turn that the slowing arms-out, speeding-up arms-in effect can be demonstrated several times over before friction finally has the last word.