@Paul Maybe it gives some intuition if you regard the flat case first. (I think this was the historical beginning of Dirac-Operators): In the flat case the difference $D^2-\nabla^*\nabla $ ist just $0$. If you now regard a bundle $S$ with curvature instead it is not too surprising that curvature comes into play. Why the curvature term decomposes into twisting- and scalar curvature might become a bit clearer if you calculate the clifford action of the induced curvature two form. Roe does this calculation in "elliplic operators, topology and asymtotic methods"(p.46 Lemma 3.13 and 3.15)

I seem not to be able to comment, so this is not an answer to the question:

@Paul Maybe it gives some intuition if you regard the flat case first. (I think this was the historical beginning of Dirac-Operators): In the flat case the difference $D^2-\nabla^*\nabla $ ist just $0$. If you now regard a bundle $S$ with curvature instead it is not too surprising that curvature comes into play. Why the curvature term decomposes into twisting- and scalar curvature might become a bit clearer if you calculate the clifford action of the induced curvature two form. Roe does this calculation in "elliplic operators, topology and asymtotic methods"(p.46 Lemma 3.13 and 3.15)