Cross Correlation Between Input-Output Sine Waves
These formulas are discrete fourier transform.

It's very standard procedure when you analyze wave-alike processes. What it does it transforms the signal from the time-space to the frequencies-space. The idea is that you take sin-wave with specific frequency, multiply it by signal and if the value is big then you have something similar to the wave of this frequency.

$\frac{1}{NT}$ term is just overall time of the sequence (N samples with length T)

The $I_{s}$ and $I_{c}$ is amplitude(intensity) for the sin and cos, respectively. You get them for the specific frequency $\omega$ and it's a single number for each. From this you can get full amplitude (G) and phase $\phi$. Why do you need phase? Because it's important - if you have two waves with similar frequency and $\phi = \pi$, then they would cancel each other, and if they have same phase, they would double.