So to start of with, you don’t actually sense temperature. What you feel is heat flux, which is the amount of energy that flows threw an area in a given time. If you walk into a room that’s been kept at 70°F overnight, the air will be pleasant, but any metal will feel cool despite being the same temperature. This is because metal is a much better at moving thermal energy (heat) than air.
So there are 4 main ways to move thermal energy. Mass loss/evaporative cooling by sweating. Conduction is when heat flows from one object to the other by the objects touching (holding an ice cube in your hand). Covection is when heat flows from an object to a moving fluid (or vise versa). And radiation is when the object throws away some of that heat energy as a photon (light).
So when your outside on a sunny day, there all 3 modes of heat transfer are in effect. Conduction, convection are most likely cooling you off due to being warmer than the environment, while radiation is heating you up. Your body likes to stay at a constant temperature, so it sweats to the point where you maintain it, thus you have no net heat flux.
When a cloud moves in front of the sun, the amount of heat added by radiation drops drastically. The air temperature didn’t change, so the heat transfer by conduction stays the same. The wind speed (along with temperature) didn’t change, so the heat transfer by convection stays the same. Your body takes some time to adjust, so for short time scales, heat tranfer by mass loss/evaporative cooling stays the same.
What this means is suddenly your operating at a net loss of heat, and that’s what you feel.
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