Published literature is not always the best place to find all the terms fully explained.
I haven't read the paper but presumably the authors are aiming to perform inference on the parameters $a$ by considering the distribution $p(y|a)$, now for whatever reason this distribution is presumably intractable in it's given form, but if we introduce some latent variables $s \in \mathcal{S} $ and the vector $W$ is some instance of $s$,
Now since $a$ is kept fixed throughout this I am going to drop it from my notation and then reintroduce it at the end, now conventional variational inference tells us that
\begin{align*}
\ln p(y) = \mathcal{L} + D_{KL},
\end{align*}
where
\begin{align*}
\mathcal{L} = \sum_s q(s) \log \frac{ p(y ,s) }{q(s)}
\end{align*}
and
\begin{align*}
D_{KL} = -\sum_s q(s) \log \frac{ p(s|y) }{q(s)}.
\end{align*}
So that using
\begin{align*}
p(y,s) = p(y|s)q(s),
\end{align*}
we have
\begin{align*}
\ln p(y) &= \sum_s q(s) \log \frac{p(y,s)}{q(s)}+ D_{KL} \\
&= \sum_s q(s) \log \frac{ p(y|s)q(s) }{q(s)} + D_{KL}\\
&= \sum_s q(s) \log p(y | s) + D_{KL} \\
&= L_s + D_{KL}
\end{align*}
Now since $D_{KL} \geq 0$ we have $L_s \leq \log p(y)$ which is the sense in which it is a "lower bound" on the log probability. To complete the conversion to their notation just add the additional conditional dependence on $a$.
Now to maximise the marginal log-likelihood for a fixed value of $a$ we can proceed to try and make $L_s$ as large as possible. Now I'm finding their notation a bit clunky - perhaps it is clearer in the article, but it seems $W$ is just a particular instance of the state of $s$, but anyway we have using the product rule and switching to the logarithmic derivative
\begin{align*}
\frac{\partial}{\partial W}L_s &= \sum_s \frac{\partial p(s|a)}{\partial W}\log p(y | s , a) + p(s|a) \frac{\partial p(y | s ,a)}{\partial W} \\
&= \sum_s \left( p(s | a)\frac{\partial \log p(s|a)}{\partial W} \right)\log p(y|s , a) + p(s |a)\frac{\partial \log p(y |s,a)}{\partial W} \\
&= \sum_s p(s|a) \left[ \frac{\partial \log p(s|a)}{\partial W}\log p(y| s,a) + \frac{\partial \log p(y| s, a )}{\partial W} \right]
\end{align*}
