It's pretty easy to model to a first approximation. Gases like Carbon Dioxide reflect certain wavelengths of infrared. When they are present in the atmosphere, they are present at a certain percentage concentration, and as that increases they reflect more infrared.

This matters because there are three ways for a body to eliminate thermal energy: radiation, convection, and conduction. Earth is not in contact with anything else, so it cannot conduct. It does not share an atmosphere with anything else, so it cannot convect. So that leaves radiation, which is largely in the form of infrared. Then it stands to reason that if you change how much IR is reflected back to the surface by the atmosphere, say by changing the amount of CO2 in the atmosphere, then you will increase the amount of heat the Earth contains as long as there is a source of heat nearby or within. Earth has both in the form of a molten core and the Sun.

This is why they call them "greenhouse gases." Because they cause the Earth to retain more heat than it radiates in the infrared spectrum, like the glass on a greenhouse.