Imagine I have a set of measures and I want to model my data as a multivariate normal distribution. For every measure I have a different mean vector and covariance matrix. So, how can I be sure about what are the parameters that fit better my data? Do I have to compute the log-likelihood for every set of parameters and see when this is minimum?
You can't compute log-likelihood's for each pair of $(\mu, \Sigma)$ since both of those parameters are real numbers, so both can take infinite number of possible values (even if you had multiple discrete-valued parameters, it might not be possible to use brute-force search in a limited time). Assuming that you don't have a closed form solution for finding the optimal parameters, otherwise you would need an optimizer (see Wikipedia article on optimization), i.e. a black-box function that takes your function and returns an optimal set of parameters.