In short: our nerves feel thermal energy enter or leave our body thus we feel temperature sensation when something nitnskin temperature interacts with us.
From what I understand: we feel the rate of heat transfer into or out of our body.
Objects feel cold or warm to us when they are at a different temperature than your skin. As a result of the ‘0th’ law of thermodynamics (heat transfer goes from high temp to low temp) there is a heat flux (energy transfer per area) that trips your nerves. How hot or cold it feels depends on the magnitude of that heat flux (depends on area of contact, thermal conductivity, temperature difference, and a couple others).
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