Rather than talking about sine and cosine waves, they motivate the Fourier transform entirely in terms of polynomials. Imagine you want to multiply two polynomials (p(x) and q(x)). The key is to recognize that there are two ways to represent each polynomial:
1. "Coefficient form," as a set of coefficients [p_0, p_1, p_2, ..., p_d] where p(x) = p_0 + p_1x + p_2x^2 + ... + p_dx^d, OR
2. "Sample form," as a set of sampled points from each polynomial, like [(0, p(0)), (1, p(1)), (2, p(2)), ..., (d, p(d))]
Now, naive multiplication of p(x) and q(x) in coefficient form takes O(d^2) scalar multiplications to get the coefficients of p(x)q(x). But if you have p(x) and q(x) in sample form, it's clear that the sample form of p(x)q(x) is just [(0, p(0)q(0)), (1, p(1)q(1)), ...], which requires only O(d) multiplications!
As long as you have enough sample points relative to the degree, these two representations are equivalent (two points uniquely defines a line, three a quadratic, four a cubic, etc.). The (inverse) Fourier transform is just a function that witnesses this equivalence, i.e., maps from representation (1) to representation (2) (and vice-versa). If the sample points are chosen cleverly (not just 1/2/3/...) it actually becomes possible to compute the Fourier transform in O(d log d) time with a DP-style algorithm (the FFT).
So, long story short, if you want to multiply p(x) and q(x), it's best to first convert them to "sample" form (O(d log d) time using the FFT), then multiply the sample forms pointwise to get the sample form of p(x)q(x) (O(d) time), and then finally convert them back to the "coefficient" form (O(d log d) using the inverse FFT).