The blanket insulates your body heat so it doesn’t escape, so you warm up. By sticking your arm or leg out, you provide a surface for heat to radiate off of you.
Your blood, as it moves quickly through your body, will go through your arm and cool down, then it returns to your body and cools off everything else as heat is dumped into it to equalize the temperature.
Which is kinda how radiators work on spacecraft, which I think is neat.
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