Carbon dioxide absorbs infra-red light. Infra-red photons are just right for exciting the atomic bonds within the carbon dioxide molecule. After absorbing an infra-red photon, the carbon dioxide molecule vibrates. The molecule is now hotter. That's what heat is: vibration of atoms.
Carbon dioxide does not reflect infra-red light. Nor does it scatter infra-red light. It absorbs infra-red light and warms up.
Our atmosphere contains carbon dioxide, roughly 0.04% by volume. This carbon dioxide absorbs infra-red radiation from the earth and the sun. Some calculations suggest that the column of air above each square meter of the earth absorbs of order 2 W of infra-red radiation from the earth. If we assume that the atmosphere is, on average, losing heat as fast as it is gaining heat, then we must ask ourselves how this 2 W is leaving the atmosphere.
The 2 W might leave by convection, in which warmer air rises. But there is nowhere for the air to go once it gets to the top of the atmosphere, so convection cannot remove the heat from the atmosphere.
The 2 W might leave by conduction, in which molecules collide with their neighbors. But air is a thermal insulator, and a 1-km column of air with a 10°C temperature difference from one end to the other will conduct only 0.0002 W of heat for every square meter of its cross-sectional area.
By elimination, the 2 W must leave the atmosphere by radiation, in which a vibrating atom emits a photon and so loses heat. This photon may be absorbed by the atmosphere before it travels more than a few hundred meters. It may be absorbed by the earth. Or it may escape the atmosphere and propagate into outer space, in which case its absorption will be greatly delayed.
So, heat leaves the atmosphere almost entirely by radiation.
PS. I am going somewhere with this argument, but I'm hoping to build up some suspense along the way.