Let's look at the master equation again:

\$\Gamma_{\sigma}'(t) = H(\Gamma_{\sigma}(t))\$

The linear operator \$$H\$$ is just matrix multiplication by \$$$H_{i,j}$\$$.

Instantiating to t=0, we get:

\$\Gamma_{\sigma}'(0) = H(\Gamma_{\sigma}(0)) = H(\sigma)\$

For a definite state \$$j\$$, this gives us:

\$\Gamma_{j}'(0) = H(\Gamma_{j}(0)) = H(j)\$

which equals the jth column of \$$$H_{i,j}$\$$.