|MadSci Network: Earth Sciences|
You are a close observer. The rate of rise and fall in temperature (in an otherwise unchanging air mass) is indeed related to the relative humidity. Dry air warms up and cools off at a faster rate than does air near saturation.
The key to understanding why this is happens is a physical concept called latent heat. Temperature is related to molecular motion. Changes in temperature correspond to the speeding up or slowing down of molecules within a parcel of air, effecting greater and lesser rates of collisions. Changes in state, on the other hand, mark a large change in molecular motion that does not correspond to a measureable change in temperature. The energy required to effect the jump from a liquid to gaseous state is known latent heat of evaporation. Molecules of gaseous water vapor store up this energy and hold it until they condense, when they release it as latent heat of condensation.
As moisture levels in a layer of the atmosphere approach saturation, gaseous water vapor molecules begin to condense into liquid form. Each molecule that condenses releases its tiny allotment of latent heat which then warms the air parcel. Multiplied over millions and millions of molecules this heating can prevent the temperature from lowering further as long as there is plenty of moisture to keep the condensation/evaporation process favoring the condensation side(the requirement that the air mass be essentially unchanging). Even below saturation levels, molecules are continuously evaporating and condensing. Once saturation is reached, the condensing side becomes dominant.
Latent heat plays an important role in many atmospheric process, most notably convection. As air rises in a convective cell, some of the moisture condenses (forming a cloud) and releases it's latent heat. This in turn fuels additional lifting. This process drives the buildup of convective clouds and thunderstorm development. This USATODAY page has a good explanation of evaporation and condensation in the atmosphere. The National Severe Storms Lab has a basic description of the role of latent heat in the development of thunderstorms. The Why Files has a section that discusses latent heat release in the development of tornados, and hurricanes.
Try the links in the MadSci Library for more information on Earth Sciences.