For the same reason a fan cools you down, even though it's not actually doing any cooling.
When you sit still, your skin warms the air around it to its own temperature. When a fan blows air at you, this warm air gets replaced with the cooler room air, and for as long as the fan blows on you, you won't be able to warm up the immediate air.
Similarly, sitting in a room about 20°C (sorry, I can't do Fahrenheit

), you're
actually sitting in a cloud of warm air several degrees hotter - or, more importantly, closer to your skin's temperature. When you jump into a bath, the water won't be at the
cloud's temperature. Hence, you experience a temperature drop.
That said, I like my baths to be a toasty 30-35°C (about 95F)