Why does (atmospheric) pressure have an effect on solubility? The Wikipedia article titled Solubility table states that 


  The table below provides information on the variation of solubility of different substances (mostly inorganic compounds) in water with temperature, at 1 atmosphere pressure.


Why does pressure have an effect of on solubility?
It comes from a combination of Henry's law and Le Châtelier's principle. Henry's law is
$$p=k_hc$$
where $p$ is the partial pressure of the gas, $k_h$ is the Henry's law constant and $c$ is the concentration of the gas (because liquids and solids are not affected as much as gases, if at all). This means that with a greater pressure comes a greater concentration, and with a lower pressure comes a lesser concentration, and vice versa

Le Châtelier's principle states that if the pressure, concentration, temperature or volume of a system changes, something else will change to compensate and swing the whole thing back into equilibrium. If there is a greater pressure, there will therefore be a grater concentration. A greater concentration means there is a greater solubility.

The UC Davis Chem Wiki (I didn't see that DrMoishe Pippik had linked to it until after I posted this!) is rather helpful - much more helpful than Wikipedia, in this case.