The "Keeling Curve" of CO₂ concentrations as measured at the Mauna Loa Observatory in Hawaii.
Source: http://www.research.noaa.gov/spotlite/archive/images/jimomedal_graph.gif

Thus, it is clear that as levels of CO₂ in the atmosphere have increased since the dawn of the industrial revolution, so too have temperatures. But we all know that correlation does not imply causation unless a reasonable forcing mechanism is proposed. So what is this forcing mechanism? This is known as the Greenhouse Effect. So how does it work?

Diagram of the Greenhouse Effect.
Source: http://www.environment.sa.gov.au/sustainability/images/greenhouse_effect.jpg

Earth absorbs energy from sunlight entering the atmosphere and emits energy out to space in the form of infrared rays. Both the incoming sunlight energy and the outgoing infrared energy are forms of radiation. However, the incoming sunlight, which is radiation with a shorter wavelength, travels unobstructed through the atmosphere (unless hitting a cloud) because this shortwave radiation is at a wavelength that is not in the absorption range of atmospheric gases. Thus, almost all of the energy of the incoming sunlight is abosrbed at the surface. On the other hand, the outgoing radiation emitted by the surface is at a wavelength that is in the absorption range of many atmospheric gases, including carbon dioxide, methane, and water vapor.
In the exact same way that, for example, if you try to shine a light through a blanket, much of the light radiation (light rays) gets blocked (because light rays are at wavelengths that are absorbed by the blanket), infrared radiation gets blocked by a carbon dioxide "blanket." Thus, the growing layer of CO₂ in our atmosphere acts as an invisible blanket that traps an increasing amount of energy in our atmosphere by absorbing and then reradiating infrared radiation back into the atmosphere. And as we increase carbon dioxide concentrations, we thicken this invisible blanket.

The Water Vapor Feedback: Carbon Dioxide can't do it alone

The greenhouse effect due to carbon dioxide is important, but it is not powerful enough on its own to cause the global warming that we are experiencing. However, a warmer climate results in an increase in water vapor in the atmosphere, as warmer air can "hold" more water vapor than colder air before reaching saturation and becoming a cloud droplet (we say "hold" for simplification, but in reality a higher air temperature at the ocean surface shifts the dynamic equilibrium between liquid water and water vapor--i.e. at all times liquid water molecules are evaporating into water vapor and water vapor molecules are condensing into liquid water--and so more water vapor is present in the air). This relationship is quantified by the Clausius-Clapeyron equation illustrated in the graph below:

Clausius-Clapeyron relationship between air temperature and water vapor "holding" capacity.
Source: http://asd-www.larc.nasa.gov/edu_act/clouds_fig28.gif

Water vapor is a powerful greenhouse gas primarily due to its abundance in the atmosphere. So increasing CO₂ increases temperature, which then increases atmospheric water vapor, which then further increases the greenhouse effect. This is known as a positive feedback.

So we have a reasonable forcing mechanism. However, this does not necessarily mean that it is the correct one. Do we have other evidence as well that would lead us to believe that this theory is correct? Yes, see Climate History.