The graph of a multivalued function on $\mathbb{C}$, such as $\log(z)$, is a surface in 4 dimensions. We can visualize it by plotting its real and imaginary parts, as done here. This link says that the imaginary part is a "faithful representation" of $\log(z)$.
What is a faithful representation of a Riemann surface?
(A cursory google search turns up no results.)
This is not a standard term at all. It appears that it is just intended to mean that the graph of the imaginary part is naturally in bijection with (or more strongly, diffeomorphic to) the Riemann surface of $\log(z)$. In other words, the projection from the 4-dimensional plot of $\log(z)$ down to the 3-dimensional plot of just its imaginary part does not collapse any points together.