The important thing is that how dry the air feels is based on how much water the air has in it as a percentage of how much it can hold (humidity as a %) and that colder air can hold less water.
Heating air is fairly straightforward. The air starts off cold and moderately humid (1 unit of water but the air can hold 2, meaning 50% humidity, a pleasant amount) and it is then heated up until it can hold much more but doesn’t have any extra water added to it (1 unit of water in the air still but now it can hold 4, giving only 25% humidity, very dry feeling).
AC takes outside air (let’s say 50% humidity with 2 units of water in the air and the air could hold 4) and cools it down, thus reducing the water capacity of it and forcing water to condense out of it (down to 1 unit of water in the air out of 1, with the other unit of water now being condensation dripping out of the system; the air is now at 100% humidity). This cold air then warms up inside your car as it fights with the warm air already in there, and this formerly cool air now has a greater capacity for water without increasing the amount of water in it, leading to a lower humidity (still 1 unit of water in the air, but now warmed up a bit so it can hold 3 units, giving only 33% humidity!).