Why does water feel cooler than air?

Firstly, to make a valid comparison between how water and air 'feels' on your skin, two conditions would need to be met:

1. Both water and air would have to be at exactly the same temperature.
2. That temperature would have to be lower than human body temperature (strictly speaking skin temperature).

If those conditions are met then water would certainly feel cooler than air. Several factors are responsible for this.

1. Water has a much higher Specific Heat Capacity than air, making it a far better coolant than air.
2. More intimate contact between water and skin, compared to air and skin, results in a higher Heat Transfer Coefficient which makes water again a better coolant.
3. In the case of fairly thin layers of water evaporative cooling also takes place in the case of water on skin. As Latent Heat of Vaporisation is carried off the water will cool down and eventually skin will cool too due to heat transfer.

Two things to consider:

1. You do not have temperature sensors sticking out into the environment. What you feel is not the temperature of the air or water; it is the temperature of the cells around the spot(s) where the temperature sensors are buried in the skin. That's why your face can "see" a hot stove element. The radiant energy from the hot metal reaches your skin and warms it slightly. As you turn your head, different skin cells get heated, allowing you to localize the hot source.
2. There is a constant flow of heat energy from the body into the environment (as long as you're alive). A lot from the core, but also from each and every muscle in the arms and legs as well; the blood supply helps to move it around. Even when the outside temperature is well above body temperature, evaporative cooling can keep you at normal body temperature. A CSI at a murder at an oasis in the Sahara would need to measure how much the liver has warmed up to determine time-of-death.

So the question to answer is: how does the temperature and nature of the surrounding medium affect the temperature of the adjacent skin cells?