This is due to the fact that skin is the interface where heat is lost.
Our body due to constant functioning, produces heat constantly as a by-product (due to exothermic reaction of ATP break mainly). The excess heat needs to be conducted away from the body, or it will cause a decrease in the body metabolism to prevent temperature rise.
Heat is lost mainly through the skin by:
- Sweating - Through evaporation
- Radiation - As heat waves (IR rays - That's why IR camera captures people at night)
- Conduction - Directly through objects that touch skin
- Convection - Through air circulation
When the ambient temperature rises, the heat lost through radiation, conduction, and convection drastically decreases. And often when the temperature is high, there is a accompanying rise in the relative humidity which decreases the heat loss through sweating (as the amount of water vapor is high in the atmosphere, the sweat does not evaporate, so no heat is lost).
So the heat which is not lost is felt as the "hot sensation". It relieves by stopping any activity, seeking shade or a cool place, etc... all of which increases the heat lost or decreases the heat produced.
You have to note that the temperature of skin is lower than the body temperature.
The Skin temperature is lower than the core body temperature for two reasons:
The skin acts as a medium through which the external temperature is measured - as such the skin temperature is at equilibrium with the external temperature. The brain regulates the core-body temperature in response to temperature measured through skin. If a person is exposed suddenly to cold environment, skin looses much of its heat in the form of radiation (radiation is direclty proportional to the temperature difference) this will cause perception of cold and the body starts shivering even though no actual heat loss has occured from the core body thermal load (only skin looses heat, not the body core). The brain anticipates that the core will loose its heat when exposed to such low temperature for prolonged periods and starts the warming mechanism before the actual cooling occurs such that the cooling is either prevented or minimized. This is called Anticipatory control and the temperature of skin being close to the ambient temperature within physiological limits is needed for this.
Skin is the medium (almost the only medium) through which the excessive heat produced by the body core during activity is expelled. The skin temperature is lower so that a constant gradient can be created between the body core and body surface to maintain the flow of heat. (Heat losses through urine and feces is minimal)
For more details see this question. All the answers in this question are good and will increase your understanding of what actually happens.