The increasing in degrees of freedom are related to an increase of the heat capacity ratio $$\gamma = \frac{c_P}{c_V},$$ but not necessarily with an increase in the internal energy of the molecule. The CO2 acts as a greenhouse effect gas because it's much more efficient than NO2 and O2 at absorbing infrarred (IR) radiation:
* The Sun has a surface temperature of ~5800 K, so most of its radiation is in the visible range of the spectrum. All the mentioned gases are more or less transparent to this type of radiation, so it goes through the atmosphere and reaches the surface.
* The surface reflects some of this radiation, which again passes through the atmosphere and goes into space, and absorbs the rest. This absorbed energy heats the surface up to ~300 K.
* Once heated, the surface emits its own radiation. However, most of this radiation is emited in IR because the temperature is much lower than the one in the Sun. NO2 and O2 are "transparent" to IR radiation as well, but CO2 is not, so it absorbs the radiation emited by the surface.
* Again the same, CO2 gets heated by absorbing radiation and emits more radiation. 184
Part of it goes again to the surface, heating it a little more, and we're back in point 3.
About the CFCs, I'm not really sure, but I'd assume they are also good IR absorbers, although their main problem is destroying the ozone layer, but that's a different topic.
Regarding degrees of freedom (DoF), you can understand them as different kinds of motion a molecule can have:
* Monoatomic: These are basically dots in the space, so only translational DoF are aviable. All translational motions can be decomposed in 3 dimensions (x,y,z), so DoF=3
* Diatomic: Two dots joined by a line. On top of the 3 translational ones, we have 2 DoF introduced by rotation (not 3 because the one with the rotation axis aligned with the atoms doesn't qualify as a movement). That makes DoF=5.
* Triatomic: here it starts to get tricky. If the three atoms are alinged like the diatomic molecule, DoF=5 again. If this is not the case (like for CO2), the extra rotation discarded has to be included, which makes DoF=6.
Pd: vibrations should be included as well, but it's very difficult to make a molecule vibrate, so it's often left aside. 104