Here's the whole scoop. Compression ratio is simply the total volume inside the cylinder when the piston is at the bottom divided by the total volume when its at the top. If its 1000cc total at the bottom and 100cc total at the top, its a 10:1 engine. Reducing the combustion chamber size by 10cc, would make it 990cc at the bottom and 90 cc at the top which is 11:1.
The compression stroke of the engine does several things like putting the piston back to the top so it has somewhere to go, concentrating the mixture around the spark plug, and adding energy to the reaction. But its more important feature is adding energy to the mixture. If you're not familiar with the idea of adiabatic heating, its when you increase the temperature of something without adding any heat. When you compress the gasses in the cylinder, they get hotter. Its the reverse of why aerosol cans feel cold when you spray on your hand. High pressure to low "spreads the heat out" making the temperature decrease. Same heat, larger volume. I'm bad at describing it, but hopefully it makes sense.
There are several kinda complicated things happening, but the higher the compression ratio, the more energy you put into the reaction. The more energy in the reaction, the more you'll get out. Bigger boom. In turn more force from the combustion can put more pressure on the piston which makes more power. It has to do with BSFC or brake specific fuel consumption which is defined as the amount of stored energy (locked up in the gasoline's hydrocarbon bonds) is released as power in the engine. Higher CRs (which can be achieved with smaller head chambers or piston crown differences) are able to release more BTUs from the gasoline as power than lower CRs.
Energy in = Energy out. In the case of compression ratio, it takes less energy to more tightly compress the gasses than you get back as extra BTUs you get released from the fuel. That equals more cylinder pressure, which equals more power.
Before somebody says it, it DOES NOT necessarily mean more MPG.
But that's a soapbox for another day.