So, I was thinking water changes state from liquid to gas at 100 degrees Celsius and becomes water vapor. So how come a glass of water left at room temperature will eventually evaporate, even though it's not hot enough?

Boiling occurs when the vapor pressure of a liquid equals the pressure of the atmosphere. Heating increases the vapor pressure. So long as the vapor pressure of the liquid is greater than the vapor pressure of that substance in the surroundings, it will evaporate. So water will evaporate quicker in dryer air and also when it is warmer.

So I guess then, my question is what causes water at room temperature to have a vapor pressure equal to that of the atmosphere?

it doesn't have a vapour pressure equal to that of the atmosphere. at room temperature it is roughly 4kPa while atmospheric is typically 101kPa.

 

it evapourates because very often the partial pressure of water in the atmosphere is less than the vapour pressure of the water. this is often reffered to as humidity.

 

if the relative humidity is 100% then the water will not evapourate as the partial pressure and the vapour pressure are in equilibrium, but if the helative humidity is lower, say 50% then the water will evapourate until it is either gone or the humidity is again at 100%