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_{x \in E} \big[ \langle z^*,y \rangle - \Psi(y) \big] \\ &=\inf_{x \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}

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}