Understanding dissolution of CO₂ in Selexol and Rectisol
Simply, $\ce{CO_{2}}$ is dissolved in those solvents in a similar way to gases dissolved in water. The molecules do not react with the solvent but mix. The higher the pressure of the gas you pump into the liquid, the more that dissolves in it (rather than turning into bubbles and escaping). In fact, amount of gases dissolved in liquids can be measured both as a concentration (mol/L) or as a partial pressure (Pa).

Henry's law states that:


  "At a constant temperature, the amount of a given gas that dissolves
  in a given type and volume of liquid is directly proportional to the
  partial pressure of that gas in equilibrium with that liquid."


The formula is:

$p = k_{H}c$

where $p$ is partial pressure of the gas in solution, $c$ is its concentration and $k_{H}$ is a constant based on what the solute is (in this case $\ce{CO_{2}}$), what the solvent is (eg methanol) and the temperature.

The harder you push, the tighter you can pack it in.

Here is a picture:



In contrast, with the amine-based solvents, the $\ce{CO_{2}}$ actually reacts, which is why a degasification process needs to happen to release it.