Stratification due to salt or heat gradients greatly affects the distribution of inert particles and living organisms in the ocean and the lower atmosphere. Laboratory studies considering the settling of a sphere in a linearly stratified fluid confirmed that stratification may dramatically enhance the drag on the body, but failed to identify the generic physical mechanism responsible for this increase. We present a rigorous splitting scheme of the various contributions to the drag on a settling body, which allows them to be properly disentangled whatever the relative magnitude of inertial, viscous, diffusive and buoyancy effects. We apply this splitting procedure to data obtained via direct numerical simulation of the flow past a settling sphere over a range of parameters covering a variety of situations of laboratory and geophysical interest. Contrary to widespread belief, we show that, in the parameter range covered by the simulations, the drag enhancement is generally not primarily due to the extra buoyancy force resulting from the dragging of light fluid by the body, but rather to the specific structure of the vorticity field set in by buoyancy effects. Simulations also reveal how the different buoyancy-induced contributions to the drag vary with the flow parameters. To unravel the origin of these variations, we analyse the different possible leading-order balances in the governing equations. Thanks to this procedure, we identify several distinct regimes which differ by the relative magnitude of length scales associated with stratification, viscosity and diffusivity. We derive the scaling laws of the buoyancy-induced drag contributions in each of these regimes. Considering tangible examples, we show how these scaling laws combined with numerical results may be used to obtain reliable predictions beyond the range of parameters covered by the simulations.