I've heard that no one 'burns fat' but rather the fat cells shrink as one consumes less calories and uses up more energy?

BUT... I have to ask how is it that the fat cells which accumulated on one's body (esp. abdomen) from years of eating more fat laden foods shrink or disappear by the act of losing weight? How does eating no fat in one's meals truly 'get rid of the fat?' Or is it simply a matter of less calories/more activity reduce the body fat?'
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What I wonder about is re those who are fat and deliberately or nonintentionally starve themselves, where does fat go then? Do those shrunken cells eventually turn into energy?
I've heard that no one 'burns fat' but rather the fat cells shrink as one consumes less calories and uses up more energy?
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