Power law rheology is of widespread occurrence in complex materials that are characterized by the presence of a very broad range of microstructural length and time scales. Although phenomenological models able to reproduce the observed rheological features exist, in general a well-established connection with the microscopic origin of this mechanical behavior is still missing. As a model system, this work focuses on a fractal colloidal gel. We thoroughly characterize the linear power law rheology of the sample and its age dependence. We show that at all sample ages and for a variety of rheological tests the gel linear viscoelasticity is very accurately described by a Fractional Maxwell (FM) model, characterized by a power law behavior. Thanks to a unique set-up that couples small-angle static and dynamic light scattering to rheological measurements, we demonstrate that the power law rheology observed in the linear regime originates from reversible non-affine rearrangements and discuss the possible relationship between the FM model and the microscopic structure of the gel.