Rain cools our bodies down because the increased humidity makes the surrounding air and our bodies colder.
As the rain water warms up it begins to evaporate therefore increasing the humidity of the air which correspondingly loses its ability to insulate which then the air its self begins to feel cooler than it is.
Just like in summer humidity makes us feel hotter the humidity in cooler temperatures makes us feel colder as well.
It basically has to do with humidity which is the main reason our bodies feel colder.