Here minimizing means decrease the distance of two distributions to its lowest: the target Bernoulli distribution and the generated result distribution. We measure the distance of two distributions using Kullback-Leibler divergence(also called relative entropy), and due to the large number theory minimizing KL divergence is amount to minimizing cross entropy(either multiclass cross entropy, see here or binary classification, see here and here).
Thus
maximizing log likelihood is equivalent to minimizing "negative log likelihood"
can be translated to
Maximizing the log likelihood is equivalent to minimizing the distance between two distributions, thus is equivalent to minimizing KL divergence, and then the cross entropy.
Maximizing the log likelihood is equivalent to minimizing the distance between two distributions, thus is equivalent to minimizing KL divergence, and then the cross entropy.
I think it has become quite intuitive.