Water evaporation

Temperature is defined as related to the average kinetic energy of the (in this case) water molecules. Since it is an average there will always be some that have a high enough velocity to penetrate the surface layer, thus escaping the liquid. This is evaporation.

 

Because the chemical potential of water is not the same for the liquid and gas phase and so the system (liquid water)+(air with water vapor) is not in a equilibrium state.
When the relative humidity is 100% it means that the chemical potential of water in the whole system is a constant. This implies that the system is at equilibrium and no more net evaporation will occur.
Here's a PDF explaining it better: http://www.inscc.utah.edu/~tgarrett/6020/Clouds/Evaporation.pdf.

 

Only the first part is actually known. The boiling point of water at atmospheric pressure is 100 C.
This first part does not imply that water evaporates only if the temperature is 100C. It evaporates at any temperature.

What is special about boiling point is that the vapor pressure becomes equal to atmospheric pressure. The visible effect is that evaporation happens in the volume (see the bubbles) and not just from the surface.

 

Water evaporates slowly at all temperatures, and the evaporation increases with water temperature increase, but the "boiling" starts at the point the water temperature reached 100 degrees Celsius at atmospheric pressure.

 

Evaporation only happens if the air is not saturated. Water is always evaporating until the air becomes saturated. Saturation vapor pressure increases with temperature. Water saturation vapor pressure at 100 C temperature is equal to 1 atmosphere. That means that at that temperature the atmosphere would have to be all water in order to saturate. At temperatures above 100 C water is trying to create a partial pressure larger than the total atmospheric pressure in order to saturate. That can't be done so water vapor rushes out trying to establish a higher atmospheric pressure so equilibrium can be reached - the poor thing. That rushing out is called boiling.