What's the mechanism behind 'losing heat through your extremities'?
Your extremities lose heat at a faster rate than the rest of your body because they have higher surface area per unit volume. Heat exchange happens at surfaces, so if you have more surface, you transfer heat more quickly. Imagine you have a sphere of water with 1L volume, and a long cylinder of water, also 1L, lets say 1cm wide, the sphere will hold heat longer because it has less surface area per unit volume.

Why does losing heat through extremities cool the rest of your body? Because of your blood. As blood runs through your core, it takes heat from surrounding tissues. The blood carries that heat to the extremities, where cooler tissues pull it away from the blood. It's kind of like how the coolant in your car takes heat from the engine block to the radiator.

When it is very cold, your body will reduce blood flow to the extremities to save heat, making your fingers vulnerable to frostbite. However, it cannot stop bloodflow, those tissues still need oxygen, so you still lose some heat.