More details based on Steve's comment: We have
\begin{align}
	-\Theta^*(-z^*) &= - \sup_{x \in E} \big[ \langle-z^*,x \rangle - \Theta(x) \big] \\
	&=\inf_{x \in E} \big[ \langle z^*,x \rangle + \Theta(x) \big]
\end{align}
and
\begin{align}
	-\Psi^*(z^*) &= - \sup_{y  \in E} \big[ \langle z^*,y \rangle - \Psi(y) \big] \\
	&=\inf_{y \in E} \big[ \langle z^*,- y \rangle + \Psi(y) \big]
\end{align}
Then,
\begin{align}
	-\Theta^*(-z^*) -\Psi^*(z^*) &=  \inf_{x,y \in E} \big[ \langle z^*,x -y\rangle + \Theta(x) + \Psi(y)\big]
\end{align}