It actually follows from the 'centre of mass' explanation. If you take the centre of mass as described of f + g you get the centre of mass of f plus the centre of mass of g. One way to explain this is to just say the + can be moved out of the integral.

Alternatively, consider the centre of mass only over the horizontal axis. Now say we only look at the 'contribution' of f + g at time t (ignoring the issues of that contribution being infinitesimal). That contribution is (f(x) + g(x)) * sin (theta) where theta is the angle of our point. Clearly this equals f(x) * sin (theta) + g(x) * sin (theta). These are the separate contributions of f(x) and g(x). The same argument holds for the centre of mass over the vertical axis (replacing sin with cos).

If we were to make the alternative explanation formal, we get back to the + being able to move outside the integral. Note that our decomposition into the horizontal and vertical part is an alternative way to de the fourier transform without complex numbers. The vertical part here is essentially the imaginary part of the fourier transform.