While the explanations involving conservation of angular momentum are perfectly accurate, I feel like a better eli5 might possibly go like this.
When you spin with legs sticking out your legs actually want to get away from you – you might feel it more if you tie or put some extra weight on your feet and try to spin up pretty fast. So when you pull your legs in while spinning you are actually doing work and expending energy. This energy has to go somewhere, so you speed up, i.e. your kinetic energy increases.
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