The first term is constant with respect to \( \boldsymbol{\Theta} \) since \( f(\boldsymbol{x}) \) is independent of \( \boldsymbol{\Theta} \). Thus the Kullback-Leibler Divergence is minimal when the second term is minimal. The second term is the log-likelihood cost function, hence minimizing the Kullback-Leibler divergence is equivalent to maximizing the log-likelihood.
To further understand generative models it is useful to study the gradient of the cost function which is needed in order to minimize it using methods like stochastic gradient descent.