Dielectric response and conduction mechanism were investigated for a multiferroic BiFe$_{0.95}$Mn$_{0.05}$O$_3$ epitaxial thin film. A contribution from a thermally activated interface (0.37 eV) and the bulk of the film on the dielectric response were observed through the comparison between experimental results and equivalent circuit model. The low frequency interface relaxation signatures strongly suggest a Maxwell-Wagner space charge origin. The alternative current conductivity deduced from the model follows a power law frequency dependence suggesting a polaronic hopping mechanism while the low frequency limit is in perfect agreement with the direct current conduction mechanism. The current-voltage characteristics were indeed correlated with Schottky-Simmons interface limited transport with activation energy of 0.36 eV, close to the one extracted from the impedance analysis. Such analysis of the electrostatic landscape and dielectric behaviour may help to further understanding the anomalous photo-induced properties in the BiFeO$_3$ system.