Answer by Zhaoting Wei for Proof of the Hirzebruch-Riemann-Roch theorem using the Atiyah-Singer index theorem

I think the top of page 148 computes the case where $W$ is trivial, whose second term $\frac{1}{2}\sum_k(Rw_k,\bar{w}_k)$ is exactly $\frac{1}{2}\text{Tr}_{T^{1,0}M}(R^+)$. The general case then follows.