Hopefully I remember this well. My adviser explained this computation to me  I don't even want to think how many years ago.

The deformation complex  of the SD equation  is $\DeclareMathOperator{\Ad}{Ad}$

$$L=d_A^-\oplus d_A^*:\Omega^1\big(\, Ad(P)\,\big)\to\Omega^2_-\big(\; \Ad(P)\;\big)\oplus \Omega^0\big(\;\Ad(P)\;\big). $$

The dimension is the index of this operator. $\DeclareMathOperator{\ind}{ind}$ $\DeclareMathOperator{\ch}{ch}$ $\DeclareMathOperator{\hA}{\widehat{A}}$This operator  obtained  by twisting with $\Ad(P)$  the   operator

$$ D=d^-+d^*:\Omega^1(M)\to \Omega^2_-(M)\oplus \Omega^0(M) $$

This is the operator $D: \Gamma(V_+\oplus V-)\to \Gamma(V_-\oplus V_-)$ in the paper you mentioned.

The Atiyah-Singer index theory  shows  that $\ind L$ is 

$$\ind L= \int_M \big[\; \ch(Ad(P)) \hA(X)\ch(V_-)\;\big]_4, $$  

where $[--]_4$ denotes the degree $4$  part of a non-homogeneous differential form.

We deduce 



$$\ch(\Ad(P))=\dim G +\ch_2(\Ad(P))+\cdots = \dim G-p_1(\Ad)+\cdots, $$

$$\ind L= \int_M (-p_1(\Ad(P))+\dim G\rho_D)  $$

where  the degree $4$ from $\rho_D= [\hA(X)\ch(V_-)]-4$ is the index density of $D$ appearinnfg in the Atiyah-Singer index theorem 
$$
\ind D=\int_M \rho_D.
$$

Thus 

$$
\ind L=-\int_M p_1(\Ad(P))+\dim G\ind D= -\int_M p_1(\Ad(P))+\dim G(b_1 -b_2^--b_0). $$

Now express $(b_2^+-b_1-b_0)$ in terms of the signature $\tau=b_2^+-b_2^-$ and the Euler characteristic $\chi=2b_0-2b_1+b_2^++b_2^-$.