Mom Rabett asked Eli to explain the greenhouse effect and why increasing CO2 increases the global temperature. Now Mom, the last surviving first grade dragon, could teach a stone to read, but she was the reason the New Math didn't work so Eli had to think really hard (and thanks for the help to the bunnies).
Mom, he said, the Earth absorbs energy from the sun and reaches a steady temperature when it radiates the same average amount of energy to space. Scientists would want Eli to insert per unit time here, and that should be understood in everything that follows.
Each part of the Earth's surface emits heat in the form of infrared (IR) radiation. The peak of this emission is right at the frequency where CO2 absorbs strongly. While the proportion of CO2 in the atmosphere is small, 380 parts per million or 0.038%, this is still a large number of molecules, large enough that near the surface, at wavelengths where CO2 absorbs, the average distance light will travel before being captured is a few meters (a couple of yards).
Greenhouse gases, as well as absorbing IR radiation, emit it. It gets a bit complicated because almost none of the greenhouse gas molecules that absorb IR light emit it immediately. Instead the internal excited energy of the molecule is transformed into thermal motion of the molecules nearby through collisions. This takes about a microsecond, a millionth of a second and is roughly a million times more likely than the molecule directly emitting IR light.
In the same way unexcited greenhouse gas molecules can be excited by collisions into a state where they emit. It turns out that the rate at which excited molecules can form and their emission spectrum is determined by the temperature, so by looking at the spectrum we can tell the level at which the Earth radiates to space.
The distance that the emitted radiation can travel is short near the surface, but increases as one climbs through the atmosphere because density, pressure and temperature decrease as we climb. Each of these lengthens the distance radiation emitted from a molecule travels before being absorbed, until about at 10 km altitude where the temperature is -50 C (or ~-60 F or~220 K) and the density has decreased by a factor of ~3, it becomes possible for radiation from CO2 molecules to reach space, carrying thermal energy away from the Earth. Below that level, energy emitted by a CO2 molecule is soon absorbed by another relatively nearby one. Thus this energy simply cannot be radiated to space to balance the incoming solar energy.
Decreasing temperature slows down the rate at which each molecule can emit while decreasing density means there are fewer greenhouse gas molecules available to absorb or radiate the energy.
Taken together this means that the doorway to space is very narrow at wavelengths where CO2 can absorb. Since the same amount of energy has to be radiated to space as is coming from the sun, something has to increase, and that is the temperature of the surface. As the surface warms, the rate at which it can radiate energy increases, pushing more thermal IR radiation out into space.
If we increase the proportion of CO2 in the atmosphere, the altitude at which energy can be radiated to space rises also, but since this higher level is colder and the pressure and density are lower, the doorway becomes narrower, and the surface has to warm more in order to shove the same amount of energy out and restore the balance with the incoming energy carried by the sunlight.
UPDATED: The first figure above shows the IR emission spectrum of the earth. The labels show where CO2 and stratospheric ozone emit. The small lines are emission from water vapor and methane and N2O absorb roughly where the step at 1250 cm-1 is. Superimposed on the spectrum are a series blackbody curves typical of different temperatures in the atmosphere. By looking at where the emission lines cross the blackbody curves we can estimate the effective temperature, and thus the altitude, at which each molecule emits to space.
For example we see that most of the CO2 emits at about 215 K, several kilometers up in the atmosphere, however the temperature of the ozone emission is about 270 K, which is that higher still in the stratosphere, about 25 km up. Finally we see the temperature of the surface is about 290 K in this spectrum in the atmospheric window where the absorption of water vapor (the small lines) is weak. By looking at where the water vapor lines meet the blackbody curves and matching the temperature to the altitude it is clear that there is not much water vapor about a couple of kilometers. That makes sense, most clouds are in the lower part of the troposphere
For those who want more detail (and some math, but lots of pretty figures) Chris Colose has something to offer and SOD has a lot more math