According to http://www.epa.gov/climatechange/kids/greenhouse.html The greenhouse effect is the process in which the emission of infrared radiation by the atmosphere warms a planet's surface. The name comes from an incorrect analogy with the warming of air inside a greenhouse compared to the air outside the greenhouse. The greenhouse effect was discovered by Joseph Fourier in 1824 and first investigated quantitatively by Svante Arrhenius in 1896.[1]
The Earth's average surface temperature of 15 °C (59 °F) is about 33 °C (59 °F) warmer than it would be without the greenhouse effect.[2] Global warming, a recent warming of the Earth's lower atmosphere, is believed to be the result of an enhanced greenhouse effect due to increased concentrations of greenhouse gases in the atmosphere. In addition to the Earth, Mars and Venus have greenhouse effects.

So why is this a natural disaster if we can't see it or necessarily feel its effects?

The average global temperature has increased by almost 1 degree F over the past century; scientists expect the average global temperature to increase an additional 2 to 6ºF over the next one hundred years. This may not sound like much, but it could change the Earth's climate as never before. At the peak of the last ice age (18,000 years ago), the temperature was only 7ºF colder than it is today, and glaciers covered much of North America!Even a small increase in temperature over a long time can change the climate.