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