The temperature drop in the sauna is simple: As you pour water on the rocks, you are consuming energy. There is energy required to heat the water and then the heat of vaporization required to turn the water into steam.
If your sauna has a bucket with a typical 100ml ladle and you sprinkle that onto the rocks. Water has a heat of vaporization of 2.3 MJ/kg, so vaporizing that ladle of water is going to take about 230 kJ. Let's say the process takes one second, so the heating of the water is going to extract 230kW from the sauna.
The sauna heater doesn't have the capacity to keep up, so energy will be extracted from stored energy in the rocks on the sauna stove and from the air of the sauna, lowering the temperature in the sauna.
The temperature will rise back to the set point of the sauna room as the heater controller increases its power output to compensate for the indicated drop in temperature. This will occur at a rate proportional to the heater power in the sauna stove, plus the thermal diffusivity constant of the sauna rocks as heat from the center of the rocks diffuses to the surface of the rocks to equalize the lost temperature from vaporizing the water.
The increased feeling of heat comes from a few other phenomena.
First, you've just added water to the sauna environment. Your body is trying to actively cool itself by evaporating water from your skin. You've just increased the partial pressure of water vapor in the system, slowing the rate of your sweat's evaporation.
Next, water vapor is better able to transfer heat than dry air. So while the air itself might be slightly cooler, you are getting more heat transferred to you both via convection and from radiation of the hot air to your body.
Third, and most significant, you are one of the coolest objects in the room, so the excess water vapor in the air is going to condense on your body, transferring 2.7 kJ to you as each gram of water condenses on your skin.
For more information see:
http://saunascape.com/2011/06/how-much-water-can-i-toss-on-the-sauna-rocks/