Without getting into the specific math, horsepower is basically determined by both torque and how fast that torque is allowed to be applied. Torque (in this case) is the force that chemical combustion creates on the pistons, which is determined by things like compression ratios (how wide the combustion chambers are, how tall or short they are, types of pistons, etc.) and combustion efficiency. Combustion efficiency is determined by how thoroughly the relevant air and fuel molecules bond before being ignited, which is aided through proper compression. This is where stoichiometric balance comes in.
Stoichiometric balance in a combustion engine is the proper mixture of air and fuel volume, specifically oxygen and hydrocarbon content, necessary for optimum combustion. The amount of fuel that is fed to the chambers will depend on how oxygenated the air that is drawn in is. But keep in mind that there is only so fast that air volume can be drawn in so the more oxygenated (cold) the air is, the faster fuel can be delivered, which will allow faster combustion. The faster combustion allows the torque to be applied at a faster rate.
The cold air intake will increase the rate of air-fuel mixture drawn in because: 1. It's more oxygenated than when it's otherwise stationed close to the hot engine and 2. It's usually designed to allow more air in anyway, regardless of greater oxygen content.
Keep in mind that combustion also produces chemical waste and, if the rate at which air-fuel mixture is drawn in increases, the emission of its waste production will necessarily need to increase as well.
So, to sum up, horsepower will be determined by how much torque is applied to create the rotational force and how fast it's allowed to be applied. Because of this, many, many things determine horsepower. The pressure in a combustion chamber will vary by actual air-fuel mixture but it's at its best with stoichiometric balance. A cold air-intake improves rate of torque by allowing the appropriate mixture to be drawn in, combusted and emitted at a faster rate.