When you step out a shower in a warm bathroom why do you shiver?

First, it's unlikely that the bathroom is warmer than the shower.

Even if it was, the hot water flowing in the shower can provide a lot of heat to your skin. The water can be quite a bit warmer than the surrounding air, and the heat transfer capacity of a thin film of flowing water is very high.

Once you turn the water off and step out of the shower there is strong evaporative cooling of the water on your skin. It's trying to come to equilibrium with its new surroundings, which is almost certainly much lower than a comfortable skin temperature.

The temperature approached is the wet bulb temperature, which is somewhere between the dry bulb and the dew point.