Through the greenhouse effect, of course.
The connection is usually made via the use of a climate model. A climate model is usually a kind of the atmospheric general circulation models which simulate the physical and dynamical processes of the atmosphere (and oceans in some models) by solving relevant equations describing these processes.
A ‘control run’ of the model is performed first and the results are regarded as a reference. Hopefully the reference results agree with observations to a certain extent.
Then other runs, in which the amounts of CO2 assumed in the model are increased (e.g., 2X or 4X), are performed and the results are compared with the reference.
If the new results show warmer surface condition than the control, then it is plausible to suspect that the increasing CO2 concentration in the atmosphere may cause warming.