This is the differential form of the Reilly formula. It holds for a function on any pseudo-Riemannian manifold. (Robert C. Reilly. Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26 (1977), no. 3, 459–472, doi:10.1512/iumj.1977.26.26036)
Use the product rule to say $$(\Delta f)^2=\operatorname{div}(\Delta f\cdot\nabla f)-\langle\nabla f,\nabla\Delta f\rangle.$$ Use the commutation formula for covariant derivatives to replace the last term by $$\langle\nabla f,\nabla\Delta f\rangle=\langle\nabla f,\Delta\nabla f\rangle-\operatorname{Ric}(\nabla f,\nabla f).$$ Use the product rule to replace the second to last term by $$\langle\nabla f,\Delta\nabla f\rangle=\operatorname{div}\big(\nabla^2f(\nabla f,\cdot)\big)-|\nabla\nabla f|^2.$$ Finally $\nabla^2f(\nabla f,\cdot)=\frac{1}{2}\nabla|\nabla f|^2$. This gives your formula.
