You mentioned CR, but what we are really talking about here is cylinder pressure, and it's simple arithmetic. Our Engines take a given amount of cylinder volume at the bottom of the stroke, essentially at atmospheric pressure, and compress it to 1/8 the volume. This is about the same as increasing the pressure by a factor of 8. When U have forced induction (Turbo or SC), U increase the pressure of what's coming in - it's no longer at atmospheric pressure - and then U increase that pressure by a factor of 8.
Atmospheric pressure @ sea level is ~ 15 Psig. If U boost 15 more #, now U have doubled the inlet pressure and also the outlet pressure. U have also ~ doubled the Lbs of Air + Fuel, which will roughly double your power. It will be kinda like doubling the CR, except for the fact that when U compress the Air/Fuel mixture U do heat it up a little bit, which will take away some of your increased mass of power mixture. That's why U add an Intercooler to try to limit this takeaway.
This is simplified, because whole books have been written on the precise details of this subject, but hopefully it explains the principle of what's going on.