Help with Markov Random Fields

I have been reading a few textbooks on MRFs. After going through all the equations, it looks like all it comes down to is minimizing an energy function and selecting some type of optimization to do that. It seems incredible simple given all the math. What am I missing?

I don't quite understand how the Gibbs distribution comes into play when it seems like you end up evaluating these energy functions which are the energy potentials. Do you end up ever actually evaluating a Gibbs distribution -exp(iU(x))?