Hot shower triggers cooling mechanisms in your body, you’ll start to sweat and your body will start moving warmer blood out to the extremities, making you feel cooler as the heat radiates/convects off. When you step out of the warm enclosure into the cooler air, your body is still trying to dump heat so you’ll feel colder.

Basically the opposite happens with a cold shower. Warm blood is kept toward your organs to keep them functioning, so your extremities are chilled by the water. On top, the shower enclosure will be colder than the rest of the room, so you move from a cold area to a warm area, so that instant temperature change makes you feel warmer.