2014CO2 is a greenhouse gas. An increase in CO2 in the atmosphere means that heat will escape to space higher up, where it is colder, which leads to less heat escaping, and hence warming.  Specifically, a CO2 increase from the pre-industrial level C0 = 280 ppmv to a concentration C will increase thermal radiation by dF W/m2 where 
dF = 5.35 * ln(C/C0)
To see how such a thermal radiation change dF affects the global mean temperature, one can look at changes in the past, for example volcanic eruptions, and see how thermal radiation and temperature correlated. These examinations, as well as others, point to a climate sensitivity such that a thermal radiation increase of 1 W/m2 will increase the temperature by around 0.8 °C. [3,4,5]
A doubling of the CO2 concentration would mean a temperature increase of about 5.35*ln(560/280)*0.8 or about 3.0 °C from the pre-industrial level, which is often how climate sensitivity is presented. A tripling gives an increase of about 4.7 °C. Here are CO2 concentrations since 1780 and the resulting temperature change, calculated using the above formula:
CDIAC and NOAA.
Following are atmospheric CO2 concentrations for the last 11,000 years:
NOAA, CDIAC and NOAA.
Projected CO2 concentrations and resulting temperature change are shown below. The CO2 projections come from the RCP 8.5 "business as usual" scenario, which, among the more well-known scenarios, has matched actual CO2 levels the closest so far, although the scenario could very well deviate later on of course, particularly on coal use perhaps.
NOAA and RCP 8.5.
Estimates of climate sensitivity, for a doubling of atmospheric CO2:
6,7,8] We then could have noted that CO2 pollution was tied to GDP,
maybe linearly (which Arrhenius maybe even did). So with exponential
economic growth (which you get from yearly growth) there could be a
problem, even though things were still unclear and there was no consensus at the time. Half a century ago things had became sufficiently clear [6,7,8] that we should have acted then.
https://www.skepticalscience.com/saturated-co2-effect-basic.htm, Dana Nuccitelli, "Is the CO2 effect saturated?"
 Gunnar Myhre et al., July 1998, "New estimates of radiative forcing due to well mixed greenhouse gases", Geophysical Research Letters 25 (14): 2715–2718, http://folk.uio.no/gunnarmy/paper/myhre_grl98.pdf
 http://www.skepticalscience.com/climate-sensitivity-advanced.htm, Dana Nuccitelli, "How sensitive is our climate?"
 Reto Knutti and Gabriele C. Hegerl, 2008-10-26, "The equilibrium sensitivity of the Earth's temperature to radiation changes", Nature Geoscience 1 (11): 735–743, https://www.ethz.ch/content/dam/ethz/special-interest/usys/iac/iac-dam/documents/group/climphys/knutti/publications/knutti08natgeo.pdf
 https://www.aip.org/history/climate/summary.htm, Spencer Weart, "The Discovery of Global Warming"
 https://www.aip.org/history/climate/index.htm, Spencer Weart, "The Discovery of Global Warming"
Daniel Vallstrom >