Well, this is RELATIVELY true. Let me explain the "why":
Fatty foods may make you feel more full, or the extra weight you gain may help add some insulation. However I haven't heard eating fatty foods would make you feel warmer. There are not even studies claiming hypothesis.
However if we look at fat from a chemical's point of view, then we know it acts as an insulator because it doesn't conduct heat as well as other types of body tissues. Ergo, having more fat would result in more insulation.
Exercise will keep you warm for a short period of time, staying hydrated will help your body function better and will work a fraction less to keep you warm. Drinking water keeps you warm because you're actually drinking H2O: oxygen for your internal combustion engine. A combustion that keeps you warm and cozy.
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