Sine (or cosine) is just the projection of uniform circular motion onto a line segment.
Why we break arbitrary periodic functions into a sum of sines and cosines (or circular motion of different integer frequencies) is because uniform circular motion is very well understood and studied, so we have many tools for working with it. It’s very easy and convenient to isolate particular terms, and each term has a pretty simple shape, and is infinitely differentiable. These bases are conveniently orthogonal relative to a uniform weight.
It's because the eigenvectors of the collection of all time shift operators are the exponential functions, and if you want to stick with real functions, then sin(ax) and cos(ax) together form a basis for the sum of the ai and -ai eigenspaces. (In the non-periodic case, you also have to deal with e^(cx) and products of this with sin and cos for all the "transients"). If you are doing something that is time-invariant, that means it commutes with time shift operators, and that means it has the same eigenvectors. Importantly, this implies such an operator can be analyzed (more or less) by how it scales these eigenvectors.
Why we break arbitrary periodic functions into a sum of sines and cosines (or circular motion of different integer frequencies) is because uniform circular motion is very well understood and studied, so we have many tools for working with it. It’s very easy and convenient to isolate particular terms, and each term has a pretty simple shape, and is infinitely differentiable. These bases are conveniently orthogonal relative to a uniform weight.