To demonstrate the derivation, we start from the full conditional distribution:
\begin{equation}\begin{split}
\rho(\mathbf{v}|\mathbf{h}) & = \frac{\rho(\mathbf{v},\mathbf{h})}{\sum_{\mathbf{v}}\rho(\mathbf{v},\mathbf{h})} \\
& =
\frac{\frac{1}{Z}\exp(\sum_jh_ja_j)\exp\big(\sum_kv^k(b_i^k + \sum_jh_jW_{i,j}^k)\big)}
{\frac{1}{Z}\exp(\sum_jh_ja_j)\sum_{\mathbf{v}}\exp\big(\sum_kv^k(b_i^k + \sum_jh_jW_{i,j}^k)\big)}
\end{split}\end{equation}
Here, the sub scripts, $i$, are dropped from the model parameters to reduce clutter. Which is fine because they in fact, index the same parameters.
\begin{equation}\begin{split} \rho(\mathbf{v}|\mathbf{h}) & =
\frac{\exp\big(\sum_k\sum_iv_i^k(b^k + \sum_jh_jW_{j}^k)\big)}
{\sum_{\mathbf{v}}\exp\big(\sum_k\sum_iv_i^k(b^k + \sum_jh_jW_{j}^k)\big)} \\
& =
\frac{\exp\big(\sum_k\sum_iv_i^k(b^k + \sum_jh_jW_{j}^k)\big)}
{\sum_{\mathbf{v}}\exp\big(\sum_k\sum_iv_i^k(b^k + \sum_jh_jW_{j}^k)\big)} \\
& = \prod\limits_{i=1}^D
\frac{\prod_{k=1}^K\exp\big(v_i^k(b^k + \sum_jh_jW_{j}^k)\big)}
{\sum_{v_i}\prod_{k=1}^K\exp\big(v_i^k(b^k + \sum_jh_jW_{j}^k)\big)}
\end{split}\end{equation}
This is the part which confused me. $v_i^k$, is a multinomial r.v. with a single trial or similarly, a categorical r.v. Therefore the sum of the denominator would be:
\begin{equation}\begin{split}
\sum_{v_i}\prod_{k=1}^K\exp\big(v_i^k(b^k + \sum_jh_jW_{j}^k)\big) &=
\exp(b^1 + \sum_jh_jW_{j}^1) +
\exp(b^2 + \sum_jh_jW_{j}^2)+ \\
& \ \ \ \ \dots +
\exp(b^K + \sum_jh_jW_{j}^K) \\
& = \sum_{q=1}^K\exp\big(b^k + \sum_jh_jW_{j}^k\big)
\end{split}\end{equation}
Hence, $\rho(\mathbf{v}|\mathbf{h})$, would be:
\begin{equation}\begin{split} \rho(\mathbf{v}|\mathbf{h}) & =
\prod\limits_{i=1}^D\prod\limits_{k=1}^K
\frac{\exp\big(v_i^k(b^k + \sum_jh_jW_{j}^k)\big)}
{\sum_{q=1}^K\exp\big(b^q + \sum_jh_jW_{j}^q\big)}
\end{split}\end{equation}
Lastly, for each i and k:
\begin{equation}\begin{split}
\rho(v_i^k|\mathbf{h}) & =
\frac{\exp\big(v_i^k(b^k + \sum_jh_jW_{j}^k)\big)}
{\sum_{q=1}^K\exp\big(b^q + \sum_jh_jW_{j}^q\big)}
\end{split}\end{equation}
